/usr/share/axiom-20170501/src/algebra/MLO.spad is in axiom-source 20170501-3.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 | )abbrev category MLO MonogenicLinearOperator
++ Author: Stephen M. Watt
++ Date Created: 1986
++ Date Last Updated: May 30, 1991
++ Description:
++ This is the category of linear operator rings with one generator.
++ The generator is not named by the category but can always be
++ constructed as \spad{monomial(1,1)}.
++
++ For convenience, call the generator \spad{G}.
++ Then each value is equal to
++ \spad{sum(a(i)*G**i, i = 0..n)}
++ for some unique \spad{n} and \spad{a(i)} in \spad{R}.
++
++ Note that multiplication is not necessarily commutative.
++ In fact, if \spad{a} is in \spad{R}, it is quite normal
++ to have \spad{a*G \^= G*a}.
MonogenicLinearOperator(R) : Category == SIG where
R : Ring
E ==> NonNegativeInteger
SIG ==> Join(Ring, BiModule(R,R)) with
if R has CommutativeRing then Algebra(R)
degree : $ -> E
++ degree(l) is \spad{n} if
++ \spad{l = sum(monomial(a(i),i), i = 0..n)}.
minimumDegree : $ -> E
++ minimumDegree(l) is the smallest \spad{k} such that
++ \spad{a(k) \^= 0} if
++ \spad{l = sum(monomial(a(i),i), i = 0..n)}.
leadingCoefficient : $ -> R
++ leadingCoefficient(l) is \spad{a(n)} if
++ \spad{l = sum(monomial(a(i),i), i = 0..n)}.
reductum : $ -> $
++ reductum(l) is \spad{l - monomial(a(n),n)} if
++ \spad{l = sum(monomial(a(i),i), i = 0..n)}.
coefficient : ($, E) -> R
++ coefficient(l,k) is \spad{a(k)} if
++ \spad{l = sum(monomial(a(i),i), i = 0..n)}.
monomial : (R, E) -> $
++ monomial(c,k) produces c times the k-th power of
++ the generating operator, \spad{monomial(1,1)}.
|