isPrimeSimple -- simple primality test of triangular systems

Synopsis

Usage:

isPrimeSimple(T)
Inputs:
- T, an instance of the type TriaSystem,
Outputs:
- a Boolean value, if the test is positive then sat(T) is prime; otherwise, it may or may not be prime

Let $T = (t_1,t_2,\cdots,t_k)$ be a triangular set (i.e., their main variables are distinct). This method verifies if the following properties hold:

(i) the main degree of $t_i$ is one for $i=1,\dots,k-1$,

(ii) $t_k$ is an irreducible polynomial.

If these properties hold then the saturated ideal of $T$ is a prime ideal.

i1 : R = QQ[x,y,z,MonomialOrder=>Lex];

i2 : F = {x*y^2 - y*z, y^3 + z^2};

i3 : T = triaSystem(R,F,{y});

i4 : isPrimeSimple(T)

o4 = true

i5 : I = saturate T

                   2            2
o5 = ideal (x*z + y , x*y - z, x  + y)

o5 : Ideal of R

i6 : isPrime I

o6 = true