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1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 | )abbrev category RPOLCAT RecursivePolynomialCategory
++ Author: Marc Moreno Maza
++ Date Created: 04/22/1994
++ Date Last Updated: 14/12/1998
++ Description:
RecursivePolynomialCategory(R,E,V) : Category == SIG where
R : Ring
E : OrderedAbelianMonoidSup
V : OrderedSet
SIG ==> PolynomialCategory(R, E, V) with
mvar : $ -> V
++ \axiom{mvar(p)} returns an error if \axiom{p} belongs to
++ \axiom{R}, otherwise returns its main variable w. r. t. to the
++ total ordering on the elements in \axiom{V}.
mdeg : $ -> NonNegativeInteger
++ \axiom{mdeg(p)} returns an error if \axiom{p} is \axiom{0},
++ otherwise, if \axiom{p} belongs to \axiom{R} returns \axiom{0},
++ otherwise, returns the degree of \axiom{p} in its main variable.
init : $ -> $
++ \axiom{init(p)} returns an error if \axiom{p} belongs to
++ \axiom{R}, otherwise returns its leading coefficient, where
++ \axiom{p} is viewed as a univariate polynomial in its main
++ variable.
head : $ -> $
++ \axiom{head(p)} returns \axiom{p} if \axiom{p} belongs to
++ \axiom{R}, otherwise returns its leading term (monomial in the
++ AXIOM sense), where \axiom{p} is viewed as a univariate polynomial
++ in its main variable.
tail : $ -> $
++ \axiom{tail(p)} returns its reductum, where \axiom{p} is viewed
++ as a univariate polynomial in its main variable.
deepestTail : $ -> $
++ \axiom{deepestTail(p)} returns \axiom{0} if \axiom{p} belongs to
++ \axiom{R}, otherwise returns tail(p), if \axiom{tail(p)} belongs
++ to \axiom{R} or \axiom{mvar(tail(p)) < mvar(p)}, otherwise
++ returns \axiom{deepestTail(tail(p))}.
iteratedInitials : $ -> List $
++ \axiom{iteratedInitials(p)} returns \axiom{[]} if \axiom{p}
++ belongs to \axiom{R},
++ otherwise returns the list of the iterated initials of \axiom{p}.
deepestInitial : $ -> $
++ \axiom{deepestInitial(p)} returns an error if \axiom{p} belongs
++ to \axiom{R},
++ otherwise returns the last term of \axiom{iteratedInitials(p)}.
leadingCoefficient : ($,V) -> $
++ \axiom{leadingCoefficient(p,v)} returns the leading coefficient
++ of \axiom{p}, where \axiom{p} is viewed as A univariate
++ polynomial in \axiom{v}.
reductum : ($,V) -> $
++ \axiom{reductum(p,v)} returns the reductum of \axiom{p}, where
++ \axiom{p} is viewed as a univariate polynomial in \axiom{v}.
monic? : $ -> Boolean
++ \axiom{monic?(p)} returns false if \axiom{p} belongs to \axiom{R},
++ otherwise returns true iff \axiom{p} is monic as a univariate
++ polynomial in its main variable.
quasiMonic? : $ -> Boolean
++ \axiom{quasiMonic?(p)} returns false if \axiom{p} belongs to
++ \axiom{R}, otherwise returns true iff the initial of \axiom{p}
++ lies in the base ring \axiom{R}.
mainMonomial : $ -> $
++ \axiom{mainMonomial(p)} returns an error if \axiom{p} is
++ \axiom{O}, otherwise, if \axiom{p} belongs to \axiom{R} returns
++ \axiom{1}, otherwise, \axiom{mvar(p)} raised to the power
++ \axiom{mdeg(p)}.
leastMonomial : $ -> $
++ \axiom{leastMonomial(p)} returns an error if \axiom{p} is
++ \axiom{O}, otherwise, if \axiom{p} belongs to \axiom{R} returns
++ \axiom{1}, otherwise, the monomial of \axiom{p} with lowest
++ degree, where \axiom{p} is viewed as a univariate polynomial in
++ its main variable.
mainCoefficients : $ -> List $
++ \axiom{mainCoefficients(p)} returns an error if \axiom{p} is
++ \axiom{O}, otherwise, if \axiom{p} belongs to \axiom{R} returns
++ [p], otherwise returns the list of the coefficients of \axiom{p},
++ where \axiom{p} is viewed as a univariate polynomial in its main
++ variable.
mainMonomials : $ -> List $
++ \axiom{mainMonomials(p)} returns an error if \axiom{p} is
++ \axiom{O}, otherwise, if \axiom{p} belongs to \axiom{R} returns
++ [1], otherwise returns the list of the monomials of \axiom{p},
++ where \axiom{p} is viewed as a univariate polynomial in its main
++ variable.
RittWuCompare : ($, $) -> Union(Boolean,"failed")
++ \axiom{RittWuCompare(a,b)} returns \axiom{"failed"} if \axiom{a}
++ and \axiom{b} have same rank w.r.t.
++ Ritt and Wu Wen Tsun ordering using the refinement of Lazard,
++ otherwise returns \axiom{infRittWu?(a,b)}.
infRittWu? : ($, $) -> Boolean
++ \axiom{infRittWu?(a,b)} returns true if \axiom{a} is less than
++ \axiom{b} w.r.t. the Ritt and Wu Wen Tsun ordering using the
++ refinement of Lazard.
supRittWu? : ($, $) -> Boolean
++ \axiom{supRittWu?(a,b)} returns true if \axiom{a} is greater
++ than \axiom{b} w.r.t. the Ritt and Wu Wen Tsun ordering using the
++ refinement of Lazard.
reduced? : ($,$) -> Boolean
++ \axiom{reduced?(a,b)} returns true iff
++ \axiom{degree(a,mvar(b)) < mdeg(b)}.
reduced? : ($,List($)) -> Boolean
++ \axiom{reduced?(q,lp)} returns true iff \axiom{reduced?(q,p)}
++ holds for every \axiom{p} in \axiom{lp}.
headReduced? : ($,$) -> Boolean
++ \axiom{headReduced?(a,b)} returns true iff
++ \axiom{degree(head(a),mvar(b)) < mdeg(b)}.
headReduced? : ($,List($)) -> Boolean
++ \axiom{headReduced?(q,lp)} returns true iff
++ \axiom{headReduced?(q,p)} holds for every \axiom{p} in \axiom{lp}.
initiallyReduced? : ($,$) -> Boolean
++ \axiom{initiallyReduced?(a,b)} returns false iff there exists an
++ iterated initial of \axiom{a} which is not reduced w.r.t \axiom{b}.
initiallyReduced? : ($,List($)) -> Boolean
++ \axiom{initiallyReduced?(q,lp)} returns true iff
++ \axiom{initiallyReduced?(q,p)} holds for every \axiom{p} in
++ \axiom{lp}.
normalized? : ($,$) -> Boolean
++ \axiom{normalized?(a,b)} returns true iff \axiom{a} and its
++ iterated initials have degree zero w.r.t. the main variable of
++ \axiom{b}
normalized? : ($,List($)) -> Boolean
++ \axiom{normalized?(q,lp)} returns true iff
++ \axiom{normalized?(q,p)} holds
++ for every \axiom{p} in \axiom{lp}.
prem : ($, $) -> $
++ \axiom{prem(a,b)} computes the pseudo-remainder of \axiom{a} by
++ \axiom{b}, both viewed as univariate polynomials in the main
++ variable of \axiom{b}.
pquo : ($, $) -> $
++ \axiom{pquo(a,b)} computes the pseudo-quotient of \axiom{a} by
++ \axiom{b}, both viewed as univariate polynomials in the main
++ variable of \axiom{b}.
prem : ($, $, V) -> $
++ \axiom{prem(a,b,v)} computes the pseudo-remainder of \axiom{a}
++ by \axiom{b}, both viewed as univariate polynomials in \axiom{v}.
pquo : ($, $, V) -> $
++ \axiom{pquo(a,b,v)} computes the pseudo-quotient of \axiom{a} by
++ \axiom{b}, both viewed as univariate polynomials in \axiom{v}.
lazyPrem : ($, $) -> $
++ \axiom{lazyPrem(a,b)} returns the polynomial \axiom{r} reduced
++ w.r.t. \axiom{b} and such that \axiom{b} divides
++ \axiom{init(b)^e a - r} where \axiom{e}
++ is the number of steps of this pseudo-division.
lazyPquo : ($, $) -> $
++ \axiom{lazyPquo(a,b)} returns the polynomial \axiom{q} such that
++ \axiom{lazyPseudoDivide(a,b)} returns \axiom{[c,g,q,r]}.
lazyPrem : ($, $, V) -> $
++ \axiom{lazyPrem(a,b,v)} returns the polynomial \axiom{r}
++ reduced w.r.t. \axiom{b} viewed as univariate polynomials in the
++ variable \axiom{v} such that \axiom{b} divides
++ \axiom{init(b)^e a - r} where \axiom{e} is the number of steps of
++ this pseudo-division.
lazyPquo : ($, $, V) -> $
++ \axiom{lazyPquo(a,b,v)} returns the polynomial \axiom{q} such that
++ \axiom{lazyPseudoDivide(a,b,v)} returns \axiom{[c,g,q,r]}.
lazyPremWithDefault : ($, $) -> _
Record (coef : $, gap : NonNegativeInteger, remainder : $)
++ \axiom{lazyPremWithDefault(a,b)} returns \axiom{[c,g,r]}
++ such that \axiom{r = lazyPrem(a,b)} and
++ \axiom{(c**g)*r = prem(a,b)}.
lazyPremWithDefault : ($, $, V) -> _
Record (coef : $, gap : NonNegativeInteger, remainder : $)
++ \axiom{lazyPremWithDefault(a,b,v)} returns \axiom{[c,g,r]}
++ such that \axiom{r = lazyPrem(a,b,v)} and
++ \axiom{(c**g)*r = prem(a,b,v)}.
lazyPseudoDivide : ($,$) -> _
Record(coef:$, gap: NonNegativeInteger,quotient:$, remainder:$)
++ \axiom{lazyPseudoDivide(a,b)} returns \axiom{[c,g,q,r]}
++ such that \axiom{[c,g,r] = lazyPremWithDefault(a,b)} and
++ \axiom{q} is the pseudo-quotient computed in this lazy
++ pseudo-division.
lazyPseudoDivide : ($,$,V) -> _
Record(coef:$, gap:NonNegativeInteger, quotient:$, remainder: $)
++ \axiom{lazyPseudoDivide(a,b,v)} returns \axiom{[c,g,q,r]} such
++ that \axiom{r = lazyPrem(a,b,v)}, \axiom{(c**g)*r = prem(a,b,v)}
++ and \axiom{q} is the pseudo-quotient computed in this lazy
++ pseudo-division.
pseudoDivide : ($, $) -> Record (quotient : $, remainder : $)
++ \axiom{pseudoDivide(a,b)} computes \axiom{[pquo(a,b),prem(a,b)]},
++ both polynomials viewed as univariate polynomials in the main
++ variable of \axiom{b}, if \axiom{b} is not a constant polynomial.
monicModulo : ($, $) -> $
++ \axiom{monicModulo(a,b)} computes \axiom{a mod b}, if \axiom{b} is
++ monic as univariate polynomial in its main variable.
lazyResidueClass : ($,$) -> _
Record(polnum:$, polden:$, power:NonNegativeInteger)
++ \axiom{lazyResidueClass(a,b)} returns \axiom{[p,q,n]} where
++ \axiom{p / q**n} represents the residue class of \axiom{a}
++ modulo \axiom{b} and \axiom{p} is reduced w.r.t. \axiom{b} and
++ \axiom{q} is \axiom{init(b)}.
headReduce: ($, $) -> $
++ \axiom{headReduce(a,b)} returns a polynomial \axiom{r} such that
++ \axiom{headReduced?(r,b)} holds and there exists an integer
++ \axiom{e} such that \axiom{init(b)^e a - r} is zero modulo
++ \axiom{b}.
initiallyReduce: ($, $) -> $
++ \axiom{initiallyReduce(a,b)} returns a polynomial \axiom{r} such
++ that \axiom{initiallyReduced?(r,b)} holds and there exists an
++ integer \axiom{e} such that \axiom{init(b)^e a - r} is zero
++ modulo \axiom{b}.
if (V has ConvertibleTo(Symbol)) then
CoercibleTo(Polynomial R)
ConvertibleTo(Polynomial R)
if R has Algebra Fraction Integer then
retractIfCan : Polynomial Fraction Integer -> Union($,"failed")
++ \axiom{retractIfCan(p)} returns \axiom{p} as an element of
++ the current domain if all its variables belong to \axiom{V}.
retract : Polynomial Fraction Integer -> $
++ \axiom{retract(p)} returns \axiom{p} as an element of the
++ current domain if \axiom{retractIfCan(p)} does not return
++ "failed", otherwise an error is produced.
convert : Polynomial Fraction Integer -> $
++ \axiom{convert(p)} returns the same as \axiom{retract(p)}.
retractIfCan : Polynomial Integer -> Union($,"failed")
++ \axiom{retractIfCan(p)} returns \axiom{p} as an element of
++ the current domain if all its variables belong to \axiom{V}.
retract : Polynomial Integer -> $
++ \axiom{retract(p)} returns \axiom{p} as an element of the
++ current domain if \axiom{retractIfCan(p)} does not return
++ "failed", otherwise an error is produced.
convert : Polynomial Integer -> $
++ \axiom{convert(p)} returns the same as \axiom{retract(p)}
if not (R has QuotientFieldCategory(Integer)) then
retractIfCan : Polynomial R -> Union($,"failed")
++ \axiom{retractIfCan(p)} returns \axiom{p} as an element
++ of the current domain if all its variables belong to
++ \axiom{V}.
retract : Polynomial R -> $
++ \axiom{retract(p)} returns \axiom{p} as an element of the
++ current domain if \axiom{retractIfCan(p)} does not
++ return "failed", otherwise an error is produced.
if (R has Algebra Integer) and not(R has Algebra Fraction Integer)
then
retractIfCan : Polynomial Integer -> Union($,"failed")
++ \axiom{retractIfCan(p)} returns \axiom{p} as an element of
++ the current domain if all its variables belong to \axiom{V}.
retract : Polynomial Integer -> $
++ \axiom{retract(p)} returns \axiom{p} as an element of the
++ current domain if \axiom{retractIfCan(p)} does not return
++ "failed", otherwise an error is produced.
convert : Polynomial Integer -> $
++ \axiom{convert(p)} returns the same as \axiom{retract(p)}.
if not (R has IntegerNumberSystem) then
retractIfCan : Polynomial R -> Union($,"failed")
++ \axiom{retractIfCan(p)} returns \axiom{p} as an element
++ of the current domain if all its variables belong to
++ \axiom{V}.
retract : Polynomial R -> $
++ \axiom{retract(p)} returns \axiom{p} as an element of the
++ current domain if \axiom{retractIfCan(p)} does not
++ return "failed", otherwise an error is produced.
if not(R has Algebra Integer) and not(R has Algebra Fraction Integer)
then
retractIfCan : Polynomial R -> Union($,"failed")
++ \axiom{retractIfCan(p)} returns \axiom{p} as an element of
++ the current domain if all its variables belong to \axiom{V}.
retract : Polynomial R -> $
++ \axiom{retract(p)} returns \axiom{p} as an element of the
++ current domain if \axiom{retractIfCan(p)} does not return
++ "failed", otherwise an error is produced.
convert : Polynomial R -> $
++ \axiom{convert(p)} returns \axiom{p} as an element of the current
++ domain if all its variables belong to \axiom{V}, otherwise an
++ error is produced.
if R has RetractableTo(Integer) then
ConvertibleTo(String)
if R has IntegralDomain then
primPartElseUnitCanonical : $ -> $
++ \axiom{primPartElseUnitCanonical(p)} returns
++ \axiom{primitivePart(p)} if \axiom{R} is a gcd-domain,
++ otherwise \axiom{unitCanonical(p)}.
primPartElseUnitCanonical! : $ -> $
++ \axiom{primPartElseUnitCanonical!(p)} replaces \axiom{p}
++ by \axiom{primPartElseUnitCanonical(p)}.
exactQuotient : ($,R) -> $
++ \axiom{exactQuotient(p,r)} computes the exact quotient of
++ \axiom{p} by \axiom{r}, which is assumed to be a divisor of
++ \axiom{p}. No error is returned if this exact quotient fails!
exactQuotient! : ($,R) -> $
++ \axiom{exactQuotient!(p,r)} replaces \axiom{p} by
++ \axiom{exactQuotient(p,r)}.
exactQuotient : ($,$) -> $
++ \axiom{exactQuotient(a,b)} computes the exact quotient of
++ \axiom{a} by \axiom{b}, which is assumed to be a divisor of
++ \axiom{a}. No error is returned if this exact quotient fails!
exactQuotient! : ($,$) -> $
++ \axiom{exactQuotient!(a,b)} replaces \axiom{a} by
++ \axiom{exactQuotient(a,b)}
subResultantGcd : ($, $) -> $
++ \axiom{subResultantGcd(a,b)} computes a gcd of \axiom{a} and
++ \axiom{b} where \axiom{a} and \axiom{b} are assumed to have the
++ same main variable \axiom{v} and are viewed as univariate
++ polynomials in \axiom{v} with coefficients in the fraction
++ field of the polynomial ring generated by their other variables
++ over \axiom{R}.
extendedSubResultantGcd : ($, $) -> _
Record (gcd : $, coef1 : $, coef2 : $)
++ \axiom{extendedSubResultantGcd(a,b)} returns \axiom{[ca,cb,r]}
++ such that \axiom{r} is \axiom{subResultantGcd(a,b)} and we have
++ \axiom{ca * a + cb * cb = r} .
halfExtendedSubResultantGcd1: ($, $) -> Record (gcd : $, coef1 : $)
++ \axiom{halfExtendedSubResultantGcd1(a,b)} returns \axiom{[g,ca]}
++ if \axiom{extendedSubResultantGcd(a,b)} returns \axiom{[g,ca,cb]}
++ otherwise produces an error.
halfExtendedSubResultantGcd2: ($, $) -> Record (gcd : $, coef2 : $)
++ \axiom{halfExtendedSubResultantGcd2(a,b)} returns \axiom{[g,cb]}
++ if \axiom{extendedSubResultantGcd(a,b)} returns \axiom{[g,ca,cb]}
++ otherwise produces an error.
resultant : ($, $) -> $
++ \axiom{resultant(a,b)} computes the resultant of \axiom{a} and
++ \axiom{b} where \axiom{a} and \axiom{b} are assumed to have the
++ same main variable \axiom{v} and are viewed as univariate
++ polynomials in \axiom{v}.
subResultantChain : ($, $) -> List $
++ \axiom{subResultantChain(a,b)}, where \axiom{a} and \axiom{b}
++ are not contant polynomials with the same main variable, returns
++ the subresultant chain of \axiom{a} and \axiom{b}.
lastSubResultant: ($, $) -> $
++ \axiom{lastSubResultant(a,b)} returns the last non-zero
++ subresultant of \axiom{a} and \axiom{b} where \axiom{a} and
++ \axiom{b} are assumed to have the same main variable \axiom{v}
++ and are viewed as univariate polynomials in \axiom{v}.
LazardQuotient: ($, $, NonNegativeInteger) -> $
++ \axiom{LazardQuotient(a,b,n)} returns \axiom{a**n exquo b**(n-1)}
++ assuming that this quotient does not fail.
LazardQuotient2: ($, $, $, NonNegativeInteger) -> $
++ \axiom{LazardQuotient2(p,a,b,n)} returns
++ \axiom{(a**(n-1) * p) exquo b**(n-1)}
++ assuming that this quotient does not fail.
next_subResultant2: ($, $, $, $) -> $
++ \axiom{nextsubResultant2(p,q,z,s)} is the multivariate version
++ of the operation
++ next_sousResultant2 from PseudoRemainderSequence from
++ the \axiomType{PseudoRemainderSequence} constructor.
if R has GcdDomain then
gcd : (R,$) -> R
++ \axiom{gcd(r,p)} returns the gcd of \axiom{r} and the content
++ of \axiom{p}.
primitivePart! : $ -> $
++ \axiom{primitivePart!(p)} replaces \axiom{p} by its primitive
++ part.
mainContent : $ -> $
++ \axiom{mainContent(p)} returns the content of \axiom{p} viewed
++ as a univariate polynomial in its main variable and with
++ coefficients in the polynomial ring generated by its other
++ variables over \axiom{R}.
mainPrimitivePart : $ -> $
++ \axiom{mainPrimitivePart(p)} returns the primitive part of
++ \axiom{p} viewed as a univariate polynomial in its main
++ variable and with coefficients in the polynomial ring generated
++ by its other variables over \axiom{R}.
mainSquareFreePart : $ -> $
++ \axiom{mainSquareFreePart(p)} returns the square free part of
++ \axiom{p} viewed as a univariate polynomial in its main
++ variable and with coefficients in the polynomial ring
++ generated by its other variables over \axiom{R}.
add
O ==> OutputForm
NNI ==> NonNegativeInteger
INT ==> Integer
exactQuo : (R,R) -> R
coerce(p:$):O ==
ground? (p) => (ground(p))::O
if (((ip := init(p))) = 1)
then
if zero?((tp := tail(p)))
then
if (((dp := mdeg(p))) = 1)
then
return((mvar(p))::O)
else
return(((mvar(p))::O **$O (dp::O)))
else
if (((dp := mdeg(p))) = 1)
then
return((mvar(p))::O +$O (tp::O))
else
return(((mvar(p))::O **$O (dp::O)) +$O (tp::O))
else
if zero?((tp := tail(p)))
then
if (((dp := mdeg(p))) = 1)
then
return((ip::O) *$O (mvar(p))::O)
else
return((ip::O) *$O ((mvar(p))::O **$O (dp::O)))
else
if ((mdeg(p)) = 1)
then
return(((ip::O) *$O (mvar(p))::O) +$O (tp::O))
((ip)::O *$O ((mvar(p))::O **$O ((mdeg(p)::O))) +$O (tail(p)::O))
mvar p ==
ground?(p) => error"Error in mvar from RPOLCAT : #1 is constant."
mainVariable(p)::V
mdeg p ==
ground?(p) => 0$NNI
degree(p,mainVariable(p)::V)
init p ==
ground?(p) => error"Error in mvar from RPOLCAT : #1 is constant."
v := mainVariable(p)::V
coefficient(p,v,degree(p,v))
leadingCoefficient (p,v) ==
zero? (d := degree(p,v)) => p
coefficient(p,v,d)
head p ==
ground? p => p
v := mainVariable(p)::V
d := degree(p,v)
monomial(coefficient(p,v,d),v,d)
reductum(p,v) ==
zero? (d := degree(p,v)) => 0$$
p - monomial(coefficient(p,v,d),v,d)
tail p ==
ground? p => 0$$
p - head(p)
deepestTail p ==
ground? p => 0$$
ground? tail(p) => tail(p)
mvar(p) > mvar(tail(p)) => tail(p)
deepestTail(tail(p))
iteratedInitials p ==
ground? p => []
p := init(p)
cons(p,iteratedInitials(p))
localDeepestInitial (p : $) : $ ==
ground? p => p
localDeepestInitial init p
deepestInitial p ==
ground? p => _
error"Error in deepestInitial from RPOLCAT : #1 is constant."
localDeepestInitial init p
monic? p ==
ground? p => false
(recip(init(p))$$ case $)@Boolean
quasiMonic? p ==
ground? p => false
ground?(init(p))
mainMonomial p ==
zero? p => error"Error in mainMonomial from RPOLCAT : #1 is zero"
ground? p => 1$$
v := mainVariable(p)::V
monomial(1$$,v,degree(p,v))
leastMonomial p ==
zero? p => error"Error in leastMonomial from RPOLCAT : #1 is zero"
ground? p => 1$$
v := mainVariable(p)::V
monomial(1$$,v,minimumDegree(p,v))
mainCoefficients p ==
zero? p => error"Error in mainCoefficients from RPOLCAT : #1 is zero"
ground? p => [p]
v := mainVariable(p)::V
coefficients(univariate(p,v)@SparseUnivariatePolynomial($))
mainMonomials p ==
zero? p => error"Error in mainMonomials from RPOLCAT : #1 is zero"
ground? p => [1$$]
v := mainVariable(p)::V
lm := monomials(univariate(p,v)@SparseUnivariatePolynomial($))
[monomial(1$$,v,degree(m)) for m in lm]
RittWuCompare (a,b) ==
(ground? b and ground? a) => "failed"::Union(Boolean,"failed")
ground? b => false::Union(Boolean,"failed")
ground? a => true::Union(Boolean,"failed")
mvar(a) < mvar(b) => true::Union(Boolean,"failed")
mvar(a) > mvar(b) => false::Union(Boolean,"failed")
mdeg(a) < mdeg(b) => true::Union(Boolean,"failed")
mdeg(a) > mdeg(b) => false::Union(Boolean,"failed")
lc := RittWuCompare(init(a),init(b))
lc case Boolean => lc
RittWuCompare(tail(a),tail(b))
infRittWu? (a,b) ==
lc : Union(Boolean,"failed") := RittWuCompare(a,b)
lc case Boolean => lc::Boolean
false
supRittWu? (a,b) ==
infRittWu? (b,a)
prem (a:$, b:$) : $ ==
cP := lazyPremWithDefault (a,b)
((cP.coef) ** (cP.gap)) * cP.remainder
pquo (a:$, b:$) : $ ==
cPS := lazyPseudoDivide (a,b)
c := (cPS.coef) ** (cPS.gap)
c * cPS.quotient
prem (a:$, b:$, v:V) : $ ==
cP := lazyPremWithDefault (a,b,v)
((cP.coef) ** (cP.gap)) * cP.remainder
pquo (a:$, b:$, v:V) : $ ==
cPS := lazyPseudoDivide (a,b,v)
c := (cPS.coef) ** (cPS.gap)
c * cPS.quotient
lazyPrem (a:$, b:$) : $ ==
(not ground?(b)) and (monic?(b)) => monicModulo(a,b)
(lazyPremWithDefault (a,b)).remainder
lazyPquo (a:$, b:$) : $ ==
(lazyPseudoDivide (a,b)).quotient
lazyPrem (a:$, b:$, v:V) : $ ==
zero? b => _
error"Error in lazyPrem : ($,$,V) -> $ from RPOLCAT : #2 is zero"
ground?(b) => 0$$
(v = mvar(b)) => lazyPrem(a,b)
dbv : NNI := degree(b,v)
zero? dbv => 0$$
dav : NNI := degree(a,v)
zero? dav => a
test : INT := dav::INT - dbv
lcbv : $ := leadingCoefficient(b,v)
while not zero?(a) and not negative?(test) repeat
lcav := leadingCoefficient(a,v)
term := monomial(lcav,v,test::NNI)
a := lcbv * a - term * b
test := degree(a,v)::INT - dbv
a
lazyPquo (a:$, b:$, v:V) : $ ==
(lazyPseudoDivide (a,b,v)).quotient
headReduce (a:$,b:$) ==
ground? b => error _
"Error in headReduce : ($,$) -> Boolean from TSETCAT : #2 is constant"
ground? a => a
mvar(a) = mvar(b) => lazyPrem(a,b)
while not reduced?((ha := head a),b) repeat
lrc := lazyResidueClass(ha,b)
if zero? tail(a)
then
a := lrc.polnum
else
a := lrc.polnum + (lrc.polden)**(lrc.power) * tail(a)
a
initiallyReduce(a:$,b:$) ==
ground? b => error _
"Error in initiallyReduce : ($,$) -> Boolean from TSETCAT : #2 is constant"
ground? a => a
v := mvar(b)
mvar(a) = v => lazyPrem(a,b)
ia := a
ma := 1$$
ta := 0$$
while (not ground?(ia)) and (mvar(ia) >= mvar(b)) repeat
if (mvar(ia) = mvar(b)) and (mdeg(ia) >= mdeg(b))
then
iamodb := lazyResidueClass(ia,b)
ia := iamodb.polnum
if not zero? ta
then
ta := (iamodb.polden)**(iamodb.power) * ta
if zero? ia
then
ia := ta
ma := 1$$
ta := 0$$
else
if not ground?(ia)
then
ta := tail(ia) * ma + ta
ma := mainMonomial(ia) * ma
ia := init(ia)
ia * ma + ta
lazyPremWithDefault (a,b) ==
ground?(b) => error _
"Error in lazyPremWithDefault from RPOLCAT : #2 is constant"
ground?(a) => [1$$,0$NNI,a]
xa := mvar a
xb := mvar b
xa < xb => [1$$,0$NNI,a]
lcb : $ := init b
db : NNI := mdeg b
test : INT := degree(a,xb)::INT - db
delta : INT := max(test + 1$INT, 0$INT)
if xa = xb
then
b := tail b
while not zero?(a) and not negative?(test) repeat
term := monomial(init(a),xb,test::NNI)
a := lcb * tail(a) - term * b
delta := delta - 1$INT
test := degree(a,xb)::INT - db
else
while not zero?(a) and not negative?(test) repeat
term := monomial(leadingCoefficient(a,xb),xb,test::NNI)
a := lcb * a - term * b
delta := delta - 1$INT
test := degree(a,xb)::INT - db
[lcb, (delta::NNI), a]
lazyPremWithDefault (a,b,v) ==
zero? b => error _
"Error in lazyPremWithDefault : ($,$,V) -> $ from RPOLCAT : #2 is zero"
ground?(b) => [b,1$NNI,0$$]
(v = mvar(b)) => lazyPremWithDefault(a,b)
dbv : NNI := degree(b,v)
zero? dbv => [b,1$NNI,0$$]
dav : NNI := degree(a,v)
zero? dav => [1$$,0$NNI,a]
test : INT := dav::INT - dbv
delta : INT := max(test + 1$INT, 0$INT)
lcbv : $ := leadingCoefficient(b,v)
while not zero?(a) and not negative?(test) repeat
lcav := leadingCoefficient(a,v)
term := monomial(lcav,v,test::NNI)
a := lcbv * a - term * b
delta := delta - 1$INT
test := degree(a,v)::INT - dbv
[lcbv, (delta::NNI), a]
pseudoDivide (a,b) ==
cPS := lazyPseudoDivide (a,b)
c := (cPS.coef) ** (cPS.gap)
[c * cPS.quotient, c * cPS.remainder]
lazyPseudoDivide (a,b) ==
ground?(b) => error _
"Error in lazyPseudoDivide from RPOLCAT : #2 is constant"
ground?(a) => [1$$,0$NNI,0$$,a]
xa := mvar a
xb := mvar b
xa < xb => [1$$,0$NNI,0$$, a]
lcb : $ := init b
db : NNI := mdeg b
q := 0$$
test : INT := degree(a,xb)::INT - db
delta : INT := max(test + 1$INT, 0$INT)
if xa = xb
then
b := tail b
while not zero?(a) and not negative?(test) repeat
term := monomial(init(a),xb,test::NNI)
a := lcb * tail(a) - term * b
q := lcb * q + term
delta := delta - 1$INT
test := degree(a,xb)::INT - db
else
while not zero?(a) and not negative?(test) repeat
term := monomial(leadingCoefficient(a,xb),xb,test::NNI)
a := lcb * a - term * b
q := lcb * q + term
delta := delta - 1$INT
test := degree(a,xb)::INT - db
[lcb, (delta::NNI), q, a]
lazyPseudoDivide (a,b,v) ==
zero? b => error _
"Error in lazyPseudoDivide : ($,$,V) -> $ from RPOLCAT : #2 is zero"
ground?(b) => [b,1$NNI,a,0$$]
(v = mvar(b)) => lazyPseudoDivide(a,b)
dbv : NNI := degree(b,v)
zero? dbv => [b,1$NNI,a,0$$]
dav : NNI := degree(a,v)
zero? dav => [1$$,0$NNI,0$$, a]
test : INT := dav::INT - dbv
delta : INT := max(test + 1$INT, 0$INT)
lcbv : $ := leadingCoefficient(b,v)
q := 0$$
while not zero?(a) and not negative?(test) repeat
lcav := leadingCoefficient(a,v)
term := monomial(lcav,v,test::NNI)
a := lcbv * a - term * b
q := lcbv * q + term
delta := delta - 1$INT
test := degree(a,v)::INT - dbv
[lcbv, (delta::NNI), q, a]
monicModulo (a,b) ==
ground?(b) => error"Error in monicModulo from RPOLCAT : #2 is constant"
rec : Union($,"failed")
rec := recip((ib := init(b)))$$
(rec case "failed")@Boolean => error _
"Error in monicModulo from RPOLCAT : #2 is not monic"
ground? a => a
ib * ((lazyPremWithDefault ((rec::$) * a,(rec::$) * b)).remainder)
lazyResidueClass(a,b) ==
zero? b => [a,1$$,0$NNI]
ground? b => [0$$,1$$,0$NNI]
ground? a => [a,1$$,0$NNI]
xa := mvar a
xb := mvar b
xa < xb => [a,1$$,0$NNI]
monic?(b) => [monicModulo(a,b),1$$,0$NNI]
lcb : $ := init b
db : NNI := mdeg b
test : INT := degree(a,xb)::INT - db
pow : NNI := 0
if xa = xb
then
b := tail b
while not zero?(a) and not negative?(test) repeat
term := monomial(init(a),xb,test::NNI)
a := lcb * tail(a) - term * b
pow := pow + 1$NNI
test := degree(a,xb)::INT - db
else
while not zero?(a) and not negative?(test) repeat
term := monomial(leadingCoefficient(a,xb),xb,test::NNI)
a := lcb * a - term * b
pow := pow + 1$NNI
test := degree(a,xb)::INT - db
[a,lcb,pow]
reduced? (a:$,b:$) : Boolean ==
degree(a,mvar(b)) < mdeg(b)
reduced? (p:$, lq : List($)) : Boolean ==
ground? p => true
while (not empty? lq) and (reduced?(p, first lq)) repeat
lq := rest lq
empty? lq
headReduced? (a:$,b:$) : Boolean ==
reduced?(head(a),b)
headReduced? (p:$, lq : List($)) : Boolean ==
reduced?(head(p),lq)
initiallyReduced? (a:$,b:$) : Boolean ==
ground? b => error _
"Error in initiallyReduced? : ($,$) -> Bool. from RPOLCAT : #2 is constant"
ground?(a) => true
mvar(a) < mvar(b) => true
(mvar(a) = mvar(b)) => reduced?(a,b)
initiallyReduced?(init(a),b)
initiallyReduced? (p:$, lq : List($)) : Boolean ==
ground? p => true
while (not empty? lq) and (initiallyReduced?(p, first lq)) repeat
lq := rest lq
empty? lq
normalized?(a:$,b:$) : Boolean ==
ground? b => error _
"Error in normalized? : ($,$) -> Boolean from TSETCAT : #2 is constant"
ground? a => true
mvar(a) < mvar(b) => true
(mvar(a) = mvar(b)) => false
normalized?(init(a),b)
normalized? (p:$, lq : List($)) : Boolean ==
while (not empty? lq) and (normalized?(p, first lq)) repeat
lq := rest lq
empty? lq
if R has IntegralDomain
then
if R has EuclideanDomain
then
exactQuo(r:R,s:R):R ==
r quo$R s
else
exactQuo(r:R,s:R):R ==
(r exquo$R s)::R
exactQuotient (p:$,r:R) ==
(p exquo$$ r)::$
exactQuotient (a:$,b:$) ==
ground? b => exactQuotient(a,ground(b))
(a exquo$$ b)::$
exactQuotient! (a:$,b:$) ==
ground? b => exactQuotient!(a,ground(b))
a := (a exquo$$ b)::$
if (R has GcdDomain) and not(R has Field)
then
primPartElseUnitCanonical p ==
primitivePart p
primitivePart! p ==
zero? p => p
if ((cp := content(p)) = 1)
then
p := unitCanonical p
else
p := unitCanonical exactQuotient!(p,cp)
p
primPartElseUnitCanonical! p ==
primitivePart! p
else
primPartElseUnitCanonical p ==
unitCanonical p
primPartElseUnitCanonical! p ==
p := unitCanonical p
if R has GcdDomain
then
gcd(r:R,p:$):R ==
(r = 1) => r
zero? p => r
ground? p => gcd(r,ground(p))$R
gcd(gcd(r,init(p)),tail(p))
mainContent p ==
zero? p => p
"gcd"/mainCoefficients(p)
mainPrimitivePart p ==
zero? p => p
(unitNormal((p exquo$$ mainContent(p))::$)).canonical
mainSquareFreePart p ==
ground? p => p
v := mainVariable(p)::V
sfp : SparseUnivariatePolynomial($)
sfp := squareFreePart(univariate(p,v)@SparseUnivariatePolynomial($))
multivariate(sfp,v)
if (V has ConvertibleTo(Symbol))
then
PR ==> Polynomial R
PQ ==> Polynomial Fraction Integer
PZ ==> Polynomial Integer
IES ==> IndexedExponents(Symbol)
Q ==> Fraction Integer
Z ==> Integer
convert(p:$) : PR ==
ground? p => (ground(p)$$)::PR
v : V := mvar(p)
d : NNI := mdeg(p)
convert(init(p))@PR *$PR _
((convert(v)@Symbol)::PR)**d +$PR convert(tail(p))@PR
coerce(p:$) : PR ==
convert(p)@PR
localRetract : PR -> $
localRetractPQ : PQ -> $
localRetractPZ : PZ -> $
localRetractIfCan : PR -> Union($,"failed")
localRetractIfCanPQ : PQ -> Union($,"failed")
localRetractIfCanPZ : PZ -> Union($,"failed")
if V has Finite
then
sizeV : NNI := size()$V
lv : List Symbol
lv := _
[convert(index(i::PositiveInteger)$V)@Symbol for i in 1..sizeV]
localRetract(p : PR) : $ ==
ground? p => (ground(p)$PR)::$
mvp : Symbol := (mainVariable(p)$PR)::Symbol
d : NNI
imvp : PositiveInteger := _
(position(mvp,lv)$(List Symbol))::PositiveInteger
vimvp : V := index(imvp)$V
xvimvp,c : $
newp := 0$$
while (not zero? (d := degree(p,mvp))) repeat
c := localRetract(coefficient(p,mvp,d)$PR)
xvimvp := monomial(c,vimvp,d)$$
newp := newp +$$ xvimvp
p := p -$PR monomial(coefficient(p,mvp,d)$PR,mvp,d)$PR
newp +$$ localRetract(p)
if R has Algebra Fraction Integer
then
localRetractPQ(pq:PQ):$ ==
ground? pq => ((ground(pq)$PQ)::R)::$
mvp : Symbol := (mainVariable(pq)$PQ)::Symbol
d : NNI
imvp : PositiveInteger := _
(position(mvp,lv)$(List Symbol))::PositiveInteger
vimvp : V := index(imvp)$V
xvimvp,c : $
newp := 0$$
while (not zero? (d := degree(pq,mvp))) repeat
c := localRetractPQ(coefficient(pq,mvp,d)$PQ)
xvimvp := monomial(c,vimvp,d)$$
newp := newp +$$ xvimvp
pq := pq -$PQ monomial(coefficient(pq,mvp,d)$PQ,mvp,d)$PQ
newp +$$ localRetractPQ(pq)
if R has Algebra Integer
then
localRetractPZ(pz:PZ):$ ==
ground? pz => ((ground(pz)$PZ)::R)::$
mvp : Symbol := (mainVariable(pz)$PZ)::Symbol
d : NNI
imvp : PositiveInteger := _
(position(mvp,lv)$(List Symbol))::PositiveInteger
vimvp : V := index(imvp)$V
xvimvp,c : $
newp := 0$$
while (not zero? (d := degree(pz,mvp))) repeat
c := localRetractPZ(coefficient(pz,mvp,d)$PZ)
xvimvp := monomial(c,vimvp,d)$$
newp := newp +$$ xvimvp
pz := pz -$PZ monomial(coefficient(pz,mvp,d)$PZ,mvp,d)$PZ
newp +$$ localRetractPZ(pz)
retractable?(p:PR):Boolean ==
lvp := variables(p)$PR
while not empty? lvp and member?(first lvp,lv) repeat
lvp := rest lvp
empty? lvp
retractablePQ?(p:PQ):Boolean ==
lvp := variables(p)$PQ
while not empty? lvp and member?(first lvp,lv) repeat
lvp := rest lvp
empty? lvp
retractablePZ?(p:PZ):Boolean ==
lvp := variables(p)$PZ
while not empty? lvp and member?(first lvp,lv) repeat
lvp := rest lvp
empty? lvp
localRetractIfCan(p : PR): Union($,"failed") ==
not retractable?(p) => "failed"::Union($,"failed")
localRetract(p)::Union($,"failed")
localRetractIfCanPQ(p : PQ): Union($,"failed") ==
not retractablePQ?(p) => "failed"::Union($,"failed")
localRetractPQ(p)::Union($,"failed")
localRetractIfCanPZ(p : PZ): Union($,"failed") ==
not retractablePZ?(p) => "failed"::Union($,"failed")
localRetractPZ(p)::Union($,"failed")
if R has Algebra Fraction Integer
then
mpc2Z := MPolyCatFunctions2(Symbol,IES,IES,Z,R,PZ,PR)
mpc2Q := MPolyCatFunctions2(Symbol,IES,IES,Q,R,PQ,PR)
ZToR (z:Z):R == coerce(z)@R
QToR (q:Q):R == coerce(q)@R
PZToPR (pz:PZ):PR == map(ZToR,pz)$mpc2Z
PQToPR (pq:PQ):PR == map(QToR,pq)$mpc2Q
retract(pz:PZ) ==
rif : Union($,"failed") := retractIfCan(pz)@Union($,"failed")
(rif case "failed") => error _
"failed in retract: POLY Z -> $ from RPOLCAT"
rif::$
convert(pz:PZ) ==
retract(pz)@$
retract(pq:PQ) ==
rif : Union($,"failed") := retractIfCan(pq)@Union($,"failed")
(rif case "failed") => error _
"failed in retract: POLY Z -> $ from RPOLCAT"
rif::$
convert(pq:PQ) ==
retract(pq)@$
if not (R has QuotientFieldCategory(Integer))
then
-- the only operation to implement is
-- retractIfCan : PR -> Union($,"failed")
-- when V does not have Finite
if V has Finite
then
retractIfCan(pr:PR) ==
localRetractIfCan(pr)@Union($,"failed")
retractIfCan(pq:PQ) ==
localRetractIfCanPQ(pq)@Union($,"failed")
else
retractIfCan(pq:PQ) ==
pr : PR := PQToPR(pq)
retractIfCan(pr)@Union($,"failed")
retractIfCan(pz:PZ) ==
pr : PR := PZToPR(pz)
retractIfCan(pr)@Union($,"failed")
retract(pr:PR) ==
rif : Union($,"failed") := _
retractIfCan(pr)@Union($,"failed")
(rif case "failed") => error _
"failed in retract: POLY Z -> $ from RPOLCAT"
rif::$
convert(pr:PR) ==
retract(pr)@$
else
-- the only operation to implement is
-- retractIfCan : PQ -> Union($,"failed")
-- when V does not have Finite
mpc2ZQ := MPolyCatFunctions2(Symbol,IES,IES,Z,Q,PZ,PQ)
mpc2RQ := MPolyCatFunctions2(Symbol,IES,IES,R,Q,PR,PQ)
ZToQ(z:Z):Q == coerce(z)@Q
RToQ(r:R):Q == retract(r)@Q
PZToPQ (pz:PZ):PQ == map(ZToQ,pz)$mpc2ZQ
PRToPQ (pr:PR):PQ == map(RToQ,pr)$mpc2RQ
retractIfCan(pz:PZ) ==
pq : PQ := PZToPQ(pz)
retractIfCan(pq)@Union($,"failed")
if V has Finite
then
retractIfCan(pq:PQ) ==
localRetractIfCanPQ(pq)@Union($,"failed")
convert(pr:PR) ==
lrif : Union($,"failed") := _
localRetractIfCan(pr)@Union($,"failed")
(lrif case "failed") => error _
"failed in convert: PR->$ from RPOLCAT"
lrif::$
else
convert(pr:PR) ==
pq : PQ := PRToPQ(pr)
retract(pq)@$
if (R has Algebra Integer) and not(R has Algebra Fraction Integer)
then
mpc2Z := MPolyCatFunctions2(Symbol,IES,IES,Z,R,PZ,PR)
ZToR (z:Z):R == coerce(z)@R
PZToPR (pz:PZ):PR == map(ZToR,pz)$mpc2Z
retract(pz:PZ) ==
rif : Union($,"failed") := retractIfCan(pz)@Union($,"failed")
(rif case "failed") => error _
"failed in retract: POLY Z -> $ from RPOLCAT"
rif::$
convert(pz:PZ) ==
retract(pz)@$
if not (R has IntegerNumberSystem)
then
-- the only operation to implement is
-- retractIfCan : PR -> Union($,"failed")
-- when V does not have Finite
if V has Finite
then
retractIfCan(pr:PR) ==
localRetractIfCan(pr)@Union($,"failed")
retractIfCan(pz:PZ) ==
localRetractIfCanPZ(pz)@Union($,"failed")
else
retractIfCan(pz:PZ) ==
pr : PR := PZToPR(pz)
retractIfCan(pr)@Union($,"failed")
retract(pr:PR) ==
rif : Union($,"failed"):=retractIfCan(pr)@Union($,"failed")
(rif case "failed") => error _
"failed in retract: POLY Z -> $ from RPOLCAT"
rif::$
convert(pr:PR) ==
retract(pr)@$
else
-- the only operation to implement is
-- retractIfCan : PZ -> Union($,"failed")
-- when V does not have Finite
mpc2RZ := MPolyCatFunctions2(Symbol,IES,IES,R,Z,PR,PZ)
RToZ(r:R):Z == retract(r)@Z
PRToPZ (pr:PR):PZ == map(RToZ,pr)$mpc2RZ
if V has Finite
then
convert(pr:PR) ==
lrif : Union($,"failed") := _
localRetractIfCan(pr)@Union($,"failed")
(lrif case "failed") => error _
"failed in convert: PR->$ from RPOLCAT"
lrif::$
retractIfCan(pz:PZ) ==
localRetractIfCanPZ(pz)@Union($,"failed")
else
convert(pr:PR) ==
pz : PZ := PRToPZ(pr)
retract(pz)@$
if not(R has Algebra Integer) and not(R has Algebra Fraction Integer)
then
-- the only operation to implement is
-- retractIfCan : PR -> Union($,"failed")
if V has Finite
then
retractIfCan(pr:PR) ==
localRetractIfCan(pr)@Union($,"failed")
retract(pr:PR) ==
rif : Union($,"failed") := retractIfCan(pr)@Union($,"failed")
(rif case "failed") => error _
"failed in retract: POLY Z -> $ from RPOLCAT"
rif::$
convert(pr:PR) ==
retract(pr)@$
if (R has RetractableTo(INT))
then
convert(pol:$):String ==
ground?(pol) => convert(retract(ground(pol))@INT)@String
ipol : $ := init(pol)
vpol : V := mvar(pol)
dpol : NNI := mdeg(pol)
tpol: $ := tail(pol)
sipol,svpol,sdpol,stpol : String
if (ipol = 1)
then
sipol := empty()$String
else
if ((-ipol) = 1)
then
sipol := "-"
else
sipol := convert(ipol)@String
if not monomial?(ipol)
then
sipol := concat(["(",sipol,")*"])$String
else
sipol := concat(sipol,"*")$String
svpol := string(convert(vpol)@Symbol)
if (dpol = 1)
then
sdpol := empty()$String
else
sdpol := _
concat("**",convert(convert(dpol)@INT)@String )$String
if zero? tpol
then
stpol := empty()$String
else
if ground?(tpol)
then
n := retract(ground(tpol))@INT
if n > 0
then
stpol := concat(" +",convert(n)@String)$String
else
stpol := convert(n)@String
else
stpol := convert(tpol)@String
if _
not member?((stpol.1)::String,["+","-"])$(List String)
then
stpol := concat(" + ",stpol)$String
concat([sipol,svpol,sdpol,stpol])$String
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