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Groebner

  

IsProper

  

decide if a given algebraic system is algebraically consistent

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

IsProper(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 IsProper command decides whether a set of polynomials J with respect to the indeterminates X is algebraically consistent (that is, whether J has at least one solution over the algebraic closure of the coefficient field). This is equivalent to testing whether 1 is a member of the ideal generated by J. The zero ideal is considered proper.

• 

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.

• 

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

Examples

withGroebner:

Fx22xz+5,xy2+yz3,3y26z3+1

F:=x22xz+5,yz3+xy2,6z3+3y2+1

(1)

IsProperF

true

(2)

IsProperF,characteristic=3

false

(3)

BasisF,tdegx,y,z,characteristic=3

1

(4)

IsProperF,x,y

false

(5)

BasisF,tdegx,y

1

(6)

withPolynomialIdeals:

JF,x

J:=x,yz3+xy2,x22xz+5,6z3+3y2+1

(7)

IsProperJ

false

(8)

BasisJ,'tord'

1

(9)

See Also

Basis

IsZeroDimensional

PolynomialIdeals

Solve

 


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