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</script></head><body><div id="package-header"><ul class="links" id="page-menu"><li><a href="src/Math-CommutativeAlgebra-GroebnerBasis.html">Source</a></li><li><a href="index.html">Contents</a></li><li><a href="doc-index.html">Index</a></li></ul><p class="caption">HaskellForMaths-0.4.8: Combinatorics, group theory, commutative algebra, non-commutative algebra</p></div><div id="content"><div id="module-header"><table class="info"><tr><th>Safe Haskell</th><td>None</td></tr><tr><th>Language</th><td>Haskell98</td></tr></table><p class="caption">Math.CommutativeAlgebra.GroebnerBasis</p></div><div id="description"><p class="caption">Description</p><div class="doc"><p>A module providing an efficient implementation of the Buchberger algorithm for calculating the (reduced) Groebner basis for an ideal,
together with some straightforward applications.</p></div></div><div id="synopsis"><p id="control.syn" class="caption expander" onclick="toggleSection('syn')">Synopsis</p><ul id="section.syn" class="hide" onclick="toggleSection('syn')"><li class="src short"><a href="#v:sPoly">sPoly</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> b, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Eq.html#t:Eq">Eq</a> k) => <a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b -> <a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b -> <a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b</li><li class="src short"><a href="#v:isGB">isGB</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> b, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Eq.html#t:Eq">Eq</a> k) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b] -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Bool.html#t:Bool">Bool</a></li><li class="src short"><a href="#v:gb1">gb1</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> b, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Eq.html#t:Eq">Eq</a> k) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b]</li><li class="src short"><a href="#v:pairWith">pairWith</a> :: (a1 -> a1 -> a) -> [a1] -> [a]</li><li class="src short"><a href="#v:reduce">reduce</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> b, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b]</li><li class="src short"><a href="#v:gb2">gb2</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> b, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b]</li><li class="src short"><a href="#v:-33-">(!)</a> :: [a] -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Int.html#t:Int">Int</a> -> a</li><li class="src short"><a href="#v:gb2a">gb2a</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> b, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b]</li><li class="src short"><a href="#v:gb3">gb3</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> b, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b]</li><li class="src short"><a href="#v:gb4">gb4</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> b, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b]</li><li class="src short"><a href="#v:mergeBy">mergeBy</a> :: (a -> a -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ordering">Ordering</a>) -> [a] -> [a] -> [a]</li><li class="src short"><a href="#v:gb">gb</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m]</li><li class="src short"><a href="#v:sugar">sugar</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m2, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m1, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m) => <a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> b m1 -> <a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> b1 m2 -> m -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Int.html#t:Int">Int</a></li><li class="src short"><a href="#v:cmpNormal">cmpNormal</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> t5, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> t4) => ((t, t4), (t1, t5)) -> ((t2, t4), (t3, t5)) -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ordering">Ordering</a></li><li class="src short"><a href="#v:cmpSug">cmpSug</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> t4, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> t3, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> t2, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Num">Num</a> t2) => ((t2, t3), (t, t4)) -> ((t2, t3), (t1, t4)) -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ordering">Ordering</a></li><li class="src short"><a href="#v:memberGB">memberGB</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Eq.html#t:Eq">Eq</a> k) => <a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Bool.html#t:Bool">Bool</a></li><li class="src short"><a href="#v:memberI">memberI</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m) => <a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Bool.html#t:Bool">Bool</a></li><li class="src short"><a href="#v:sumI">sumI</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m]</li><li class="src short"><a href="#v:productI">productI</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m]</li><li class="src short"><a href="#v:intersectI">intersectI</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m]</li><li class="src short"><a href="#v:toElimFst">toElimFst</a> :: (<a href="Math-Algebras-Structures.html#t:Mon">Mon</a> b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Functor.html#t:Functor">Functor</a> f) => f a -> f (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Elim2">Elim2</a> a b)</li><li class="src short"><a href="#v:toElimSnd">toElimSnd</a> :: (<a href="Math-Algebras-Structures.html#t:Mon">Mon</a> a, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Functor.html#t:Functor">Functor</a> f) => f b -> f (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Elim2">Elim2</a> a b)</li><li class="src short"><a href="#v:isElimFst">isElimFst</a> :: (<a href="Math-Algebras-Structures.html#t:Mon">Mon</a> a, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Eq.html#t:Eq">Eq</a> a) => <a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> b (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Elim2">Elim2</a> a t) -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Bool.html#t:Bool">Bool</a></li><li class="src short"><a href="#v:fromElimSnd">fromElimSnd</a> :: <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Functor.html#t:Functor">Functor</a> f => f (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Elim2">Elim2</a> t b) -> f b</li><li class="src short"><a href="#v:eliminateFst">eliminateFst</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> t, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> b1, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> t, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> b1) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> b1 (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Elim2">Elim2</a> t b)] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> b1 b]</li><li class="src short"><a href="#v:quotientI">quotientI</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m]</li><li class="src short"><a href="#v:quotientP">quotientP</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> <a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m]</li><li class="src short"><a href="#v:eliminate">eliminate</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Eq.html#t:Eq">Eq</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:MonomialConstructor">MonomialConstructor</a> m, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> (m v), <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> (m v)) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k (m v)] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k (m v)] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k (m v)]</li><li class="src short"><a href="#v:mbasis">mbasis</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> a, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Num">Num</a> a) => [a] -> [a]</li><li class="src short"><a href="#v:mbasisQA">mbasisQA</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m]</li><li class="src short"><a href="#v:ltIdeal">ltIdeal</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m]</li><li class="src short"><a href="#v:numMonomials">numMonomials</a> :: <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Integral">Integral</a> a => a -> a -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Integer">Integer</a></li><li class="src short"><a href="#v:hilbertFunQA">hilbertFunQA</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Int.html#t:Int">Int</a> -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Integer">Integer</a></li><li class="src short"><a href="#v:hilbertSeriesQA1">hilbertSeriesQA1</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Int.html#t:Int">Int</a>]</li><li class="src short"><a href="#v:hilbertSeriesQA">hilbertSeriesQA</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Integer">Integer</a>]</li><li class="src short"><a href="#v:hilbertSeriesQA-39-">hilbertSeriesQA'</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:MonomialConstructor">MonomialConstructor</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> (m v), <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> (m v), <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k (m v)) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k (m v)] -> [<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Integer">Integer</a>]</li><li class="src short"><a href="#v:hilbertPolyQA">hilbertPolyQA</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> <a href="Math-CommutativeAlgebra-Polynomial.html#t:GlexPoly">GlexPoly</a> <a href="Math-Core-Field.html#t:Q">Q</a> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-String.html#t:String">String</a></li><li class="src short"><a href="#v:hilbertPolyQA-39-">hilbertPolyQA'</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:MonomialConstructor">MonomialConstructor</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> (m v), <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> (m v), <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k (m v)) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k (m v)] -> <a href="Math-CommutativeAlgebra-Polynomial.html#t:GlexPoly">GlexPoly</a> <a href="Math-Core-Field.html#t:Q">Q</a> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-String.html#t:String">String</a></li><li class="src short"><a href="#v:dim">dim</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Int.html#t:Int">Int</a></li><li class="src short"><a href="#v:dim-39-">dim'</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> (m v), <a href="Math-CommutativeAlgebra-Polynomial.html#t:MonomialConstructor">MonomialConstructor</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k (m v), <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> (m v), <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k (m v)] -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Int.html#t:Int">Int</a></li></ul></div><div id="interface"><h1>Documentation</h1><div class="top"><p class="src"><a id="v:sPoly" class="def">sPoly</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> b, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Eq.html#t:Eq">Eq</a> k) => <a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b -> <a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b -> <a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#sPoly" class="link">Source</a> <a href="#v:sPoly" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:isGB" class="def">isGB</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> b, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Eq.html#t:Eq">Eq</a> k) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b] -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Bool.html#t:Bool">Bool</a> <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#isGB" class="link">Source</a> <a href="#v:isGB" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:gb1" class="def">gb1</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> b, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Eq.html#t:Eq">Eq</a> k) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b] <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#gb1" class="link">Source</a> <a href="#v:gb1" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:pairWith" class="def">pairWith</a> :: (a1 -> a1 -> a) -> [a1] -> [a] <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#pairWith" class="link">Source</a> <a href="#v:pairWith" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:reduce" class="def">reduce</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> b, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b] <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#reduce" class="link">Source</a> <a href="#v:reduce" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:gb2" class="def">gb2</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> b, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b] <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#gb2" class="link">Source</a> <a href="#v:gb2" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:-33-" class="def">(!)</a> :: [a] -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Int.html#t:Int">Int</a> -> a <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#%21" class="link">Source</a> <a href="#v:-33-" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:gb2a" class="def">gb2a</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> b, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b] <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#gb2a" class="link">Source</a> <a href="#v:gb2a" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:gb3" class="def">gb3</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> b, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b] <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#gb3" class="link">Source</a> <a href="#v:gb3" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:gb4" class="def">gb4</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> b, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k b] <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#gb4" class="link">Source</a> <a href="#v:gb4" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:mergeBy" class="def">mergeBy</a> :: (a -> a -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ordering">Ordering</a>) -> [a] -> [a] -> [a] <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#mergeBy" class="link">Source</a> <a href="#v:mergeBy" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:gb" class="def">gb</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#gb" class="link">Source</a> <a href="#v:gb" class="selflink">#</a></p><div class="doc"><p>Given a list of polynomials over a field, return a Groebner basis for the ideal generated by the polynomials.</p></div></div><div class="top"><p class="src"><a id="v:sugar" class="def">sugar</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m2, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m1, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m) => <a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> b m1 -> <a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> b1 m2 -> m -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Int.html#t:Int">Int</a> <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#sugar" class="link">Source</a> <a href="#v:sugar" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:cmpNormal" class="def">cmpNormal</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> t5, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> t4) => ((t, t4), (t1, t5)) -> ((t2, t4), (t3, t5)) -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ordering">Ordering</a> <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#cmpNormal" class="link">Source</a> <a href="#v:cmpNormal" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:cmpSug" class="def">cmpSug</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> t4, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> t3, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> t2, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Num">Num</a> t2) => ((t2, t3), (t, t4)) -> ((t2, t3), (t1, t4)) -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ordering">Ordering</a> <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#cmpSug" class="link">Source</a> <a href="#v:cmpSug" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:memberGB" class="def">memberGB</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Eq.html#t:Eq">Eq</a> k) => <a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Bool.html#t:Bool">Bool</a> <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#memberGB" class="link">Source</a> <a href="#v:memberGB" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:memberI" class="def">memberI</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m) => <a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Bool.html#t:Bool">Bool</a> <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#memberI" class="link">Source</a> <a href="#v:memberI" class="selflink">#</a></p><div class="doc"><p><code>memberI f gs</code> returns whether f is in the ideal generated by gs</p></div></div><div class="top"><p class="src"><a id="v:sumI" class="def">sumI</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#sumI" class="link">Source</a> <a href="#v:sumI" class="selflink">#</a></p><div class="doc"><p>Given ideals I and J, their sum is defined as I+J = {f+g | f <- I, g <- J}.</p><p>If fs and gs are generators for I and J, then <code>sumI fs gs</code> returns generators for I+J.</p><p>The geometric interpretation is that the variety of the sum is the intersection of the varieties,
ie V(I+J) = V(I) intersect V(J)</p></div></div><div class="top"><p class="src"><a id="v:productI" class="def">productI</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#productI" class="link">Source</a> <a href="#v:productI" class="selflink">#</a></p><div class="doc"><p>Given ideals I and J, their product I.J is the ideal generated by all products {f.g | f <- I, g <- J}.</p><p>If fs and gs are generators for I and J, then <code>productI fs gs</code> returns generators for I.J.</p><p>The geometric interpretation is that the variety of the product is the union of the varieties,
ie V(I.J) = V(I) union V(J)</p></div></div><div class="top"><p class="src"><a id="v:intersectI" class="def">intersectI</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#intersectI" class="link">Source</a> <a href="#v:intersectI" class="selflink">#</a></p><div class="doc"><p>The intersection of ideals I and J is the set of all polynomials which belong to both I and J.</p><p>If fs and gs are generators for I and J, then <code>intersectI fs gs</code> returns generators for the intersection of I and J</p><p>The geometric interpretation is that the variety of the intersection is the union of the varieties,
ie V(I intersect J) = V(I) union V(J).</p><p>The reason for prefering the intersection over the product is that the intersection of radical ideals is radical,
whereas the product need not be.</p></div></div><div class="top"><p class="src"><a id="v:toElimFst" class="def">toElimFst</a> :: (<a href="Math-Algebras-Structures.html#t:Mon">Mon</a> b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Functor.html#t:Functor">Functor</a> f) => f a -> f (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Elim2">Elim2</a> a b) <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#toElimFst" class="link">Source</a> <a href="#v:toElimFst" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:toElimSnd" class="def">toElimSnd</a> :: (<a href="Math-Algebras-Structures.html#t:Mon">Mon</a> a, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Functor.html#t:Functor">Functor</a> f) => f b -> f (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Elim2">Elim2</a> a b) <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#toElimSnd" class="link">Source</a> <a href="#v:toElimSnd" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:isElimFst" class="def">isElimFst</a> :: (<a href="Math-Algebras-Structures.html#t:Mon">Mon</a> a, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Eq.html#t:Eq">Eq</a> a) => <a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> b (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Elim2">Elim2</a> a t) -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Bool.html#t:Bool">Bool</a> <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#isElimFst" class="link">Source</a> <a href="#v:isElimFst" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:fromElimSnd" class="def">fromElimSnd</a> :: <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Functor.html#t:Functor">Functor</a> f => f (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Elim2">Elim2</a> t b) -> f b <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#fromElimSnd" class="link">Source</a> <a href="#v:fromElimSnd" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:eliminateFst" class="def">eliminateFst</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> t, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> b1, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> t, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> b, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> b1) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> b1 (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Elim2">Elim2</a> t b)] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> b1 b] <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#eliminateFst" class="link">Source</a> <a href="#v:eliminateFst" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:quotientI" class="def">quotientI</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#quotientI" class="link">Source</a> <a href="#v:quotientI" class="selflink">#</a></p><div class="doc"><p>Given ideals I and J, their quotient is defined as I:J = {f | f <- R, f.g is in I for all g in J}.</p><p>If fs and gs are generators for I and J, then <code>quotientI fs gs</code> returns generators for I:J.</p><p>The ideal quotient is the algebraic analogue of the Zariski closure of a difference of varieties.
V(I:J) contains the Zariski closure of V(I)-V(J), with equality if k is algebraically closed and I is a radical ideal.</p></div></div><div class="top"><p class="src"><a id="v:quotientP" class="def">quotientP</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> <a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#quotientP" class="link">Source</a> <a href="#v:quotientP" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:eliminate" class="def">eliminate</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Eq.html#t:Eq">Eq</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:MonomialConstructor">MonomialConstructor</a> m, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> (m v), <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> (m v)) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k (m v)] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k (m v)] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k (m v)] <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#eliminate" class="link">Source</a> <a href="#v:eliminate" class="selflink">#</a></p><div class="doc"><p><code>eliminate vs gs</code> returns the elimination ideal obtained from the ideal generated by gs by eliminating the variables vs.</p></div></div><div class="top"><p class="src"><a id="v:mbasis" class="def">mbasis</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> a, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Num">Num</a> a) => [a] -> [a] <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#mbasis" class="link">Source</a> <a href="#v:mbasis" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:mbasisQA" class="def">mbasisQA</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#mbasisQA" class="link">Source</a> <a href="#v:mbasisQA" class="selflink">#</a></p><div class="doc"><p>Given variables vs, and a Groebner basis gs, <code>mbasisQA vs gs</code> returns a monomial basis for the quotient algebra k[vs]/<gs>.
For example, <code>mbasisQA [x,y] [x^2+y^2-1]</code> returns a monomial basis for k[x,y]/<x^2+y^2-1>.
In general, the monomial basis is likely to be infinite.</p></div></div><div class="top"><p class="src"><a id="v:ltIdeal" class="def">ltIdeal</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#ltIdeal" class="link">Source</a> <a href="#v:ltIdeal" class="selflink">#</a></p><div class="doc"><p>Given an ideal I, the leading term ideal lt(I) consists of the leading terms of all elements of I.
If I is generated by gs, then <code>ltIdeal gs</code> returns generators for lt(I).</p></div></div><div class="top"><p class="src"><a id="v:numMonomials" class="def">numMonomials</a> :: <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Integral">Integral</a> a => a -> a -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Integer">Integer</a> <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#numMonomials" class="link">Source</a> <a href="#v:numMonomials" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:hilbertFunQA" class="def">hilbertFunQA</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Int.html#t:Int">Int</a> -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Integer">Integer</a> <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#hilbertFunQA" class="link">Source</a> <a href="#v:hilbertFunQA" class="selflink">#</a></p><div class="doc"><p>Given variables vs, and a homogeneous ideal gs, <code>hilbertFunQA vs gs</code> returns the Hilbert function for the quotient algebra k[vs]/<gs>.
Given an integer i, the Hilbert function returns the number of degree i monomials in a basis for k[vs]/<gs>.
For a homogeneous ideal, this number is independent of the monomial ordering used
(even though the elements of the monomial basis themselves are dependent on the ordering).</p><p>If the ideal I is not homogeneous, then R/I is not graded, and the Hilbert function is not well-defined.
Specifically, the number of degree i monomials in a basis is likely to depend on which monomial ordering you use.</p></div></div><div class="top"><p class="src"><a id="v:hilbertSeriesQA1" class="def">hilbertSeriesQA1</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Int.html#t:Int">Int</a>] <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#hilbertSeriesQA1" class="link">Source</a> <a href="#v:hilbertSeriesQA1" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:hilbertSeriesQA" class="def">hilbertSeriesQA</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Integer">Integer</a>] <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#hilbertSeriesQA" class="link">Source</a> <a href="#v:hilbertSeriesQA" class="selflink">#</a></p><div class="doc"><p>Given variables vs, and a homogeneous ideal gs, <code>hilbertSeriesQA vs gs</code> returns the Hilbert series for the quotient algebra k[vs]/<gs>.
The Hilbert series should be interpreted as a formal power series where the coefficient of t^i is the Hilbert function evaluated at i.
That is, the i'th element in the series is the number of degree i monomials in a basis for k[vs]/<gs>.</p></div></div><div class="top"><p class="src"><a id="v:hilbertSeriesQA-39-" class="def">hilbertSeriesQA'</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:MonomialConstructor">MonomialConstructor</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> (m v), <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> (m v), <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k (m v)) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k (m v)] -> [<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Integer">Integer</a>] <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#hilbertSeriesQA%27" class="link">Source</a> <a href="#v:hilbertSeriesQA-39-" class="selflink">#</a></p><div class="doc"><p>In the case where every variable v occurs in some generator g of the homogeneous ideal (the usual case),
then the vs can be inferred from the gs.
<code>hilbertSeriesQA' gs</code> returns the Hilbert series for the quotient algebra k[vs]/<gs>.</p></div></div><div class="top"><p class="src"><a id="v:hilbertPolyQA" class="def">hilbertPolyQA</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> <a href="Math-CommutativeAlgebra-Polynomial.html#t:GlexPoly">GlexPoly</a> <a href="Math-Core-Field.html#t:Q">Q</a> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-String.html#t:String">String</a> <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#hilbertPolyQA" class="link">Source</a> <a href="#v:hilbertPolyQA" class="selflink">#</a></p><div class="doc"><p>For i >> 0, the Hilbert function becomes a polynomial in i, called the Hilbert polynomial.</p></div></div><div class="top"><p class="src"><a id="v:hilbertPolyQA-39-" class="def">hilbertPolyQA'</a> :: (<a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="Math-CommutativeAlgebra-Polynomial.html#t:MonomialConstructor">MonomialConstructor</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> (m v), <a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> (m v), <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k (m v)) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k (m v)] -> <a href="Math-CommutativeAlgebra-Polynomial.html#t:GlexPoly">GlexPoly</a> <a href="Math-Core-Field.html#t:Q">Q</a> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-String.html#t:String">String</a> <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#hilbertPolyQA%27" class="link">Source</a> <a href="#v:hilbertPolyQA-39-" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:dim" class="def">dim</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> m, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k m] -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Int.html#t:Int">Int</a> <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#dim" class="link">Source</a> <a href="#v:dim" class="selflink">#</a></p></div><div class="top"><p class="src"><a id="v:dim-39-" class="def">dim'</a> :: (<a href="Math-CommutativeAlgebra-Polynomial.html#t:Monomial">Monomial</a> (m v), <a href="Math-CommutativeAlgebra-Polynomial.html#t:MonomialConstructor">MonomialConstructor</a> m, <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k (m v), <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> (m v), <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Ord.html#t:Ord">Ord</a> k, <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Prelude.html#t:Fractional">Fractional</a> k) => [<a href="Math-Algebras-VectorSpace.html#t:Vect">Vect</a> k (m v)] -> <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.9.0.0/Data-Int.html#t:Int">Int</a> <a href="src/Math-CommutativeAlgebra-GroebnerBasis.html#dim%27" class="link">Source</a> <a href="#v:dim-39-" class="selflink">#</a></p></div></div></div><div id="footer"><p>Produced by <a href="http://www.haskell.org/haddock/">Haddock</a> version 2.17.2</p></div></body></html>
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