SPolynomial(f, g, T, characteristic=p)
a MonomialOrder or ShortMonomialOrder
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 lcm⁡LT⁡f,LT⁡g⁢fLT⁡f−gLT⁡g, 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.
f ≔ x−13⁢y2−12⁢z3
g ≔ x2−x⁢y+92⁢z
Operators in a Weyl algebra
A ≔ diff_algebra⁡Dx,x,Dy,y,polynom=x,y:
T ≔ MonomialOrder⁡A,tdeg⁡Dx,Dy,x,y:
Operators in a q-calculus algebra
A ≔ skew_algebra⁡comm=q,qdilat=Sx,x,q:
T ≔ MonomialOrder⁡A,tdeg⁡Sx:
Operators in a Weyl algebra modulo a prime
A ≔ diff_algebra⁡Dx,x,characteristic=2:
T ≔ MonomialOrder⁡A,tdeg⁡Dx:
Algebraic number coefficients
s ≔ SPolynomial⁡2−3⁢i⁢x2−x,x2+1+i⁢x,tdeg⁡x
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