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# isSquarefreePseudomonomialIdeal -- determine if an ideal is a square free pseudomonomial ideal

## Synopsis

• Usage:
isSquarefreePseudomonomialIdeal(I)
• Inputs:
• Outputs:
• , true or false

## Description

An ideal is a square free pseudomonomial ideal if its generators are square free pseudomonomials. A square free pseudomonomial is a polynomial of the form P=z1*z2*...*zk such that zi is either xi or xi-1 and a variable can only appear once in P. For example, x1*x3, x1*(x2-1)*(x3-1) are square free pseudomonomial and x2*x3^2, x1*(x1-1)*(x3-1) are not.
 i1 : R=ZZ/2[x1,x2,x3,x4,x5];  i2 : I = ideal(x1*x2,x3*x4,x5); o2 : Ideal of R i3 : isSquarefreePseudomonomialIdeal(I) o3 = true
 i4 : R=ZZ/2[x1,x2]; i5 : I = ideal(x1*(x2-1),(x1-1)*(x2-1),x1*x2,(x1-1)*x2); o5 : Ideal of R i6 : isSquarefreePseudomonomialIdeal(I) o6 = true
 i7 : R=ZZ/3[x1,x2]; i8 : I = ideal(x1,x2-1); o8 : Ideal of R i9 : isSquarefreePseudomonomialIdeal(I) o9 = true
 i10 : R=QQ[x1,x2,x3,x4,x5]; i11 : I = ideal(x1*(x1-1),(x3-1)*x4,x5); o11 : Ideal of R i12 : not isSquarefreePseudomonomialIdeal(I) o12 = true
 i13 : R=QQ[x1,x2,x3,x4,x5]; i14 : I = ideal(x1*(x5-1),(x3-1)*x4^2,x5); o14 : Ideal of R i15 : isSquarefreePseudomonomialIdeal(I) o15 = false

## Caveat

The algorithm disregards the base field. So, ideal(-x1) is a square free pseudomonomial ideal in F[x1] for F=ZZ,ZZ/p,QQ, whereas ideal(-x1) is not a monomial ideal when F=ZZ.

## Ways to use isSquarefreePseudomonomialIdeal :

• "isSquarefreePseudomonomialIdeal(Ideal)"

## For the programmer

The object isSquarefreePseudomonomialIdeal is .