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Groebner

  

Solve

  

factoring Buchberger algorithm

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

Solve(G, X, NZ, opts)

Parameters

G

-

a list or set of polynomials or a PolynomialIdeal

X

-

(optional) a list or set of variables or a ShortMonomialOrder

NZ

-

(optional) a list or set of non-zero constraints

opts

-

optional arguments of the form keyword=value.

Description

• 

The Solve command preprocesses an algebraic system for solving by computing a collection of reduced Groebner bases corresponding to subvarieties of the initial varieties.  First G is subdivided by factorization.  Then each component is passed to a variant of Buchberger's algorithm that factors all intermediate results.  Multiplicities are removed.

• 

The second argument X specifies the variables of the system and optionally the monomial order. If X is a list Solve(G, X) computes lexicographic Groebner bases for each component with the variables ordered according to X.  When X is a set an order is chosen heuristically (see SuggestVariableOrder) and Solve again computes lexicographic Groebner bases for each component. If X is a ShortMonomialOrder then Solve will use X for all computations, however keep in mind that lexicographic orders tend to produce the most factorization.

• 

The third argument NZ is used to prevent certain quantities from being considered in roots. If one of the polynomials in NZ vanishes identically on a component, then that component is removed. If NZ is is omitted then  is assumed.

• 

The optional argument characteristic=p can be used to specify the ring characteristic when G is a list or set of polynomials.  The default characteristic is zero.

• 

If G is a list or set of polynomials, Solve returns a set of elements each consisting of a Groebner basis, a monomial order, and a set of non-zero constraints.  If G is a PolynomialIdeal then Solve combines the Groebner basis and monomial order into a PolynomialIdeal data structure. Either format can be selected using the optional argument output=basis or output=ideal, respectively.

• 

Note that the gsolve command is deprecated.  It may not be supported in a future Maple release.

Examples

withGroebner:

Gx22xz+5,xy2+yz3,3y28z3:

SolveG,x,y,z

z,y,x2+5,plexx,y,z,,9z696z5+240z3+1600,3z5+32z440z2+80y,9z596z4+120z2+640x,plexx,y,z,z,3z532z4+40z2

(1)

SolveG,y,z,x

x2+5,z,y,plexy,z,x,,3x664x5+45x4+225x2+375,3x564x4+45x3+150x+150z,8x+3y,plexy,z,x,x,x2+5

(2)

SolveG,x,y,z,y

9z696z5+240z3+1600,3z5+32z440z2+80y,9z596z4+120z2+640x,plexx,y,z,z,3z532z4+40z2

(3)

SolveG,x,y,z,characteristic=3

z,y,x+1,plexx,y,z,,z,y,x+2,plexx,y,z,

(4)

SolveG,tdegx,y,z

z,y,x2+5,tdegx,y,z,,8x+3y,9y2+48yz+320,3z3+6yz+40,tdegx,y,z,y

(5)

Example where the multiplicities change. The point (x,y) = (0,-1) is a double point of intersection for the two curves x^2 + y^2 - 1 = 0 and y - x^2 + 1 = 0, but appears only with multiplicity one in the result.

withPolynomialIdeals:

Jx2+y21,yx2+1

J:=x2+y+1,x2+y21

(6)

SolveJ,x,y

x,y+1,,y,x1,,y,x+1,

(7)

See Also

Basis

Monomial Orders

SuggestVariableOrder

 


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