SPolynomial - Maple Help

Groebner

 SPolynomial
 compute S-polynomials

 Calling Sequence SPolynomial(f, g, T, characteristic=p)

Parameters

 f, g - polynomials T - a MonomialOrder or ShortMonomialOrder p - (optional) characteristic

Description

 • SPolynomial(f, g, T) computes an S-polynomial of f and g with respect to the monomial order T. The S-polynomial is a syzygy.  It induces a cancellation of leading terms using the smallest possible multiples of f and g.
 • In commutative domains the S-polynomial of f and g is given by $\mathrm{lcm}\left(\mathrm{LT}\left(f\right),\mathrm{LT}\left(g\right)\right)\left(\frac{f}{\mathrm{LT}\left(f\right)}-\frac{g}{\mathrm{LT}\left(g\right)}\right)$, where LT(f) denotes the leading term of f with respect to T. In case of Ore algebras the S-polynomial is defined similarly, however since there is no longer a division on monomials the S-polynomial of f and g is defined by c'[f]*t'[f]*f - c'[g]*t'[g]*g where:
 – t'[f]*LM(f) = t'[g]*LM(g) = lcm(LM(f), LM(g))  where LM(f) denotes the leading monomial of f
 – t'[f]*LC(f) = c''[f]*t'[f] + lower order terms where LC(f) denotes the leading coefficient of f
 – t'[g]*LC(g) = c''[g]*t'[g] + lower order terms
 – c'[f]*c''[f] = c'[g]*c''[g] = c''[f]*c''[g] / gcd(c''[f], c''[g])
 • An optional argument characteristic=p can be used to specify the ring characteristic when T is a ShortMonomialOrder.  The default value is zero.
 • If T is a ShortMonomialOrder then f and g must be polynomials in the ring implied by T.  If T is a MonomialOrder created with the Groebner[MonomialOrder] command, then f and g must be members of the algebra used to define T.
 • Note that the spoly command is deprecated.  It may not be supported in a future Maple release.

Examples

 > $\mathrm{with}\left(\mathrm{Groebner}\right):$
 > $f≔x-13{y}^{2}-12{z}^{3}$
 ${f}{≔}{-}{12}{}{{z}}^{{3}}{-}{13}{}{{y}}^{{2}}{+}{x}$ (1)
 > $g≔{x}^{2}-xy+92z$
 ${g}{≔}{{x}}^{{2}}{-}{x}{}{y}{+}{92}{}{z}$ (2)
 > $\mathrm{SPolynomial}\left(f,g,\mathrm{plex}\left(x,y,z\right)\right)$
 ${-}{12}{}{x}{}{{z}}^{{3}}{-}{13}{}{x}{}{{y}}^{{2}}{+}{x}{}{y}{-}{92}{}{z}$ (3)
 > $\mathrm{SPolynomial}\left(f,g,\mathrm{tdeg}\left(x,y,z\right)\right)$
 ${-}{12}{}{x}{}{y}{}{{z}}^{{3}}{-}{13}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{1104}{}{{z}}^{{4}}{+}{{x}}^{{3}}$ (4)

Operators in a Weyl algebra

 > $\mathrm{with}\left(\mathrm{Ore_algebra}\right):$
 > $A≔\mathrm{diff_algebra}\left(\left[\mathrm{Dx},x\right],\left[\mathrm{Dy},y\right],\mathrm{polynom}=\left\{x,y\right\}\right):$
 > $T≔\mathrm{MonomialOrder}\left(A,\mathrm{tdeg}\left(\mathrm{Dx},\mathrm{Dy},x,y\right)\right):$
 > $\mathrm{SPolynomial}\left(\mathrm{Dx}+y,\mathrm{Dy}-x,T\right)$
 ${\mathrm{Dx}}{}{x}{+}{\mathrm{Dy}}{}{y}{+}{2}$ (5)

Operators in a q-calculus algebra

 > $A≔\mathrm{skew_algebra}\left(\mathrm{comm}=q,\mathrm{qdilat}=\left[\mathrm{Sx},x,q\right]\right):$
 > $T≔\mathrm{MonomialOrder}\left(A,\mathrm{tdeg}\left(\mathrm{Sx}\right)\right):$
 > $\mathrm{SPolynomial}\left({\mathrm{Sx}}^{2}-x,x\mathrm{Sx},T\right)$
 ${-}{q}{}{{x}}^{{2}}$ (6)

Operators in a Weyl algebra modulo a prime

 > $A≔\mathrm{diff_algebra}\left(\left[\mathrm{Dx},x\right],\mathrm{characteristic}=2\right):$
 > $T≔\mathrm{MonomialOrder}\left(A,\mathrm{tdeg}\left(\mathrm{Dx}\right)\right):$
 > $\mathrm{SPolynomial}\left(\mathrm{Dx},{x}^{2},T\right)$
 ${0}$ (7)

Algebraic number coefficients

 > $s≔\mathrm{SPolynomial}\left(\left(2-3i\right){x}^{2}-x,{x}^{2}+\left(1+i\right)x,\mathrm{tdeg}\left(x\right)\right)$
 ${s}{≔}\left({3}{}{{i}}^{{2}}{+}{i}{-}{3}\right){}{x}$ (8)
 > $\mathrm{eval}\left(s,i=I\right)$
 $\left({-6}{+}{I}\right){}{x}$ (9)
 > $\mathrm{SPolynomial}\left(\left(2-3I\right){x}^{2}-x,{x}^{2}+\left(1+I\right)x,\mathrm{tdeg}\left(x\right)\right)$
 $\left({-6}{+}{I}\right){}{x}$ (10)