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Groebner

  

IsZeroDimensional

  

decide if a system has a finite number of solutions

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

IsZeroDimensional(J, X, characteristic=p)

Parameters

J

-

a list or set of polynomials or a PolynomialIdeal

X

-

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

p

-

(optional) characteristic

Description

• 

The IsZeroDimensional command decides whether a set of polynomials J with respect to the indeterminates X has a finite number of solutions over the algebraic closure of the coefficient field.  For example, in characteristic zero this command tests whether there are a finite number of solutions in the complex numbers. In every domain this test is equivalent to testing whether the HilbertDimension is zero.

• 

The variables of the system can be specified using an optional second argument X. If X is a ShortMonomialOrder then a Groebner basis of J with respect to X is computed. By default, X is the set of all indeterminates not appearing inside a RootOf command or radical when J is a list or set, or PolynomialIdeals[IdealInfo][Variables](J) if J is an ideal.

• 

The optional argument characteristic=p specifies the ring characteristic when J is a list or set. This option has no effect when J is a PolynomialIdeal or when X is a MonomialOrder.

• 

The algorithm for IsZeroDimensional tests whether a power of each variable appears as a leading monomial in a Groebner basis for J. To access this functionality directly (as a subroutine in your program), make J a list or set of leading monomials. IsZeroDimensional will detect this case and execute the algorithm with minimal overhead.

• 

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

Examples

withGroebner:

Fx22xz+5,xy2+yz3,3y28z3

F:=x22xz+5,yz3+xy2,8z3+3y2

(1)

IsZeroDimensionalF

true

(2)

LeadingMonomialBasisF,tdegx,y,z,tdegx,y,z

x2,z3,xy2,y4

(3)

IsZeroDimensionalF,characteristic=2

false

(4)

LeadingMonomialBasisF,tdegx,y,z,characteristic=2,tdegx,y,z

y2,x2,yz3

(5)

IsZeroDimensionalF1..2

false

(6)

HilbertDimensionF1..2

1

(7)

IsZeroDimensionalF1..2,x,y

true

(8)

withPolynomialIdeals:

JF

J:=8z3+3y2,yz3+xy2,x22xz+5

(9)

NumberOfSolutionsJ

18

(10)

NormalSetJ,tdegx,y,z1

1,z,y,x,z2,yz,xz,y2,xy,yz2,xz2,y2z,xyz,y3,y2z2,xyz2,y3z,y3z2

(11)

See Also

Basis

HilbertDimension

IsProper

NormalSet

PolynomialIdeals[NumberOfSolutions]

 


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