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.

$\mathrm{with}\left(\mathrm{Groebner}\right)\:$

$f≔x-13y^2-12z^3$

$f≔-12z^3-13y^2+x$

$g≔x^2-xy+92z$

$g≔x^2-xy+92z$

$\mathrm{SPolynomial}\left(f\,g\,\mathrm{plex}\left(x\,y\,z\right)\right)$

$-12x{z}^3-13x{y}^2+xy-92z$

$\mathrm{SPolynomial}\left(f\,g\,\mathrm{tdeg}\left(x\,y\,z\right)\right)$

$-12xy{z}^3-13x^2{y}^2+1104z^4+x^3$

$\mathrm{with}\left(\mathrm{Ore\_algebra}\right)\:$

$A≔\mathrm{diff\_algebra}\left(\left[\mathrm{Dx}\,x\right]\,\left[\mathrm{Dy}\,y\right]\,\mathrm{polynom}=\left\{x\,y\right\}\right)\:$

$T≔\mathrm{MonomialOrder}\left(A\,\mathrm{tdeg}\left(\mathrm{Dx}\,\mathrm{Dy}\,x\,y\right)\right)\:$

$\mathrm{SPolynomial}\left(\mathrm{Dx}+y\,\mathrm{Dy}-x\,T\right)$

$\mathrm{Dx}x+\mathrm{Dy}y+2$

$A≔\mathrm{skew\_algebra}\left(\mathrm{comm}=q\,\mathrm{qdilat}=\left[\mathrm{Sx}\,x\,q\right]\right)\:$

$T≔\mathrm{MonomialOrder}\left(A\,\mathrm{tdeg}\left(\mathrm{Sx}\right)\right)\:$

$\mathrm{SPolynomial}\left({\mathrm{Sx}}^2-x\,x\mathrm{Sx}\,T\right)$

$-q{x}^2$

$A≔\mathrm{diff\_algebra}\left(\left[\mathrm{Dx}\,x\right]\,\mathrm{characteristic}=2\right)\:$

$T≔\mathrm{MonomialOrder}\left(A\,\mathrm{tdeg}\left(\mathrm{Dx}\right)\right)\:$

$\mathrm{SPolynomial}\left(\mathrm{Dx}\,x^2\,T\right)$

$0$

$s≔\mathrm{SPolynomial}\left(\left(2-3i\right)x^2-x\,x^2+\left(1+i\right)x\,\mathrm{tdeg}\left(x\right)\right)$

$s≔\left(3i^2+i-3\right)x$

$\mathrm{eval}\left(s\,i=\mathrm{I}\right)$

$\left(\mathrm{-6}+\mathrm{I}\right)x$

$\mathrm{SPolynomial}\left(\left(2-3\mathrm{I}\right)x^2-x\,x^2+\left(1+\mathrm{I}\right)x\,\mathrm{tdeg}\left(x\right)\right)$

$\left(\mathrm{-6}+\mathrm{I}\right)x$