Groebner - Maple Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Algebra : Polynomials : Groebner : Groebner/SPolynomial

Groebner

  

SPolynomial

  

compute S-polynomials

 

Calling Sequence

Parameters

Description

Examples

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 lcmLTf,LTgfLTfgLTg, 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

withGroebner:

fx13y212z3

f:=12z313y2+x

(1)

gx2xy+92z

g:=x2xy+92z

(2)

SPolynomialf,g,plexx,y,z

12xz313xy2+xy92z

(3)

SPolynomialf,g,tdegx,y,z

12xyz313x2y2+1104z4+x3

(4)

Operators in a Weyl algebra

withOre_algebra:

Adiff_algebraDx,x,Dy,y,polynom=x,y:

TMonomialOrderA,tdegDx,Dy,x,y:

SPolynomialDx+y,Dyx,T

Dxx+Dyy+2

(5)

Operators in a q-calculus algebra

Askew_algebracomm=q,qdilat=Sx,x,q:

TMonomialOrderA,tdegSx:

SPolynomialSx2x,xSx,T

qx2

(6)

Operators in a Weyl algebra modulo a prime

Adiff_algebraDx,x,characteristic=2:

TMonomialOrderA,tdegDx:

SPolynomialDx,x2,T

0

(7)

Algebraic number coefficients

sSPolynomial23ix2x,x2+1+ix,tdegx

s:=3i2+i3x

(8)

si=I|si=I

6+Ix

(9)

SPolynomial23Ix2x,x2+1+Ix,tdegx

6+Ix

(10)

See Also

Basis

MonomialOrder

Ore_algebra

 


Download Help Document

Was this information helpful?



Please add your Comment (Optional)
E-mail Address (Optional)
What is ? This question helps us to combat spam