
278 Algebraic Operads: An Algorithmic Companion
takes as input positive integers k, m, n and whose output is a pseudorandom
m ×n matrix whose entries are chosen uniformly from E.
Exercise 8.5. Write a computer program which takes as input an m × n
matrix A whose entries belong to the polynomial ring F[x
1
, . . . , x
k
] and which
produces as output Gröbner bases for the determinantal ideals DI
r
(A) where
0 ≤ r ≤ min(m, n). Extend this to a program which also computes the zero
sets V (DI
r
(A)) and the inverse image for each possible rank:
inverse image rank(s)
V (DI
r+1
(A) \V (DI
r
(A)) 0 ≤ r < min(m, n)
F
k
\ V (DI
r
(A)) r = min(m, n)
Exercise 8.6. Verify the claims ...