Degree of the minimum polynomial

Problem : be sure that the degree is correct (d over the integers)

for a random prime pi:

either minpoly mod pi has degree d

or pi divides some d ? d non-zero minor of A

but then there are a few possible such bad primes

• Choose primes from a sufficiently big set

• Select polynomials mod pi with maximal degree

  ex: for homology matrices with 3030 primes between 215 and 216 :
  
  • at most 1,7 % of “bad” primes
  • with 10 polynomials of same degree : 2.10-16 probability of failure

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