primaryDecomposition issue with nonstandard grading

[eloisagrifo]

input:

R = QQ[x,y,z, Degrees => {25,72,29}];
P = ker map(QQ[t],R,{t^(25),t^(72),t^(29)});
primaryDecomposition(P^3)

output:

[stdio:3:20:(3)](https://www.unimelb-macaulay2.cloud.edu.au/#editor:stdio:3:20:(3)): error: array index 0 out of bounds 0 .. -1

This used to work as recently as version 1.16, but it's been broken for the last few versions of M2, and it is still broken in version 1.23.0.1. There was a discussion about this in the google groups in March 2022 and the conclusion seemed to be that the issue is with the Birational strategy in minimal primes.

[d-torrance]

Reproducing this with errorDepth set to 2, we get:

i4 : primaryDecomposition (P^3)
/usr/share/Macaulay2/MinimalPrimes/AnnotatedIdeal.m2:202:34:(2):[56]: error: array index 0 out of bounds 0 .. -1
/usr/share/Macaulay2/MinimalPrimes/AnnotatedIdeal.m2:202:34:(2):[56]: --entering debugger (type help to see debugger commands)
/usr/share/Macaulay2/MinimalPrimes/AnnotatedIdeal.m2:202:18-202:34: --source code:
        S := ring I.LexGBOverBase#0; -- should be of the form kk(indepvars)[fibervars]

ii5 : peek I

oo5 = AnnotatedIdeal{Birational => true                                         }
                     DecomposeMonomials => true
                     Factorization => true
                     IndependentSet => ({z}, QQ[y, x, z], frac(QQ[z])[y, x])
                     isPrime => YES
                     LexGBOverBase => {}
                     LexGBSplit => true
                     Linear => true
                     Linears => {}
                     NonzeroDivisors => {}
                                      3    4 4   29     18   7     7 2 14    25
                     Ideal => ideal (y  - x z , x   - 3x  y*z  + 3x y z   - z  )
                                   11      7
                     Inverted => {x   - y*z , z}

So the bug seems to be that I.LexGBOverbase is an empty list, but we're trying to access its 0th element.

[mikestillman]

This same thing just occurred on an integral closure bug that @eisenbud gave me. I'm looking into this bug. Not sure how it got introduced. This is deep in the decompose code!