Presented By: Department of Mathematics
Commutative Algebra
Small subalgebras of polynomial rings and Stillman's conjecture in large characteristic for all degrees
This is the first of two talks on joint work of Tigran Ananyan and the speaker. The main result is that there is a positive integer B(n,d) such that given n forms F_i of positive degree at most d in a polynomial ring R in N variables over an algebraically closed field K of characteristic 0 or greater than d (note that B = B(n,d) does NOT depend on N or K), there exists a regular sequence of forms g_1, ..., g_h, where h is at most B, of degree at most d such that the F_i are in the ring K[g_1, ..., g_h]. This places the F_i in a "small" polynomial subring of R over which R is faithfully flat (in fact, free). This implies Stillman's conjecture that the projective dimension of the ideal I generated by the F_i is bounded independent of N in the case where the characteristic is 0 or large: it will be at most B(n,d). The authors had
earlier shown this only up to degree 4. One can also bound all numerical data about the
primary decomposition of I independent of N. The authors have also obtained these
results for d = 2 and d = 3 with no restriction on the characteristic, and in degree 4 except in characteristic 2. Many open questions remain. Speaker(s): Mel Hochster (University of Michigan)
earlier shown this only up to degree 4. One can also bound all numerical data about the
primary decomposition of I independent of N. The authors have also obtained these
results for d = 2 and d = 3 with no restriction on the characteristic, and in degree 4 except in characteristic 2. Many open questions remain. Speaker(s): Mel Hochster (University of Michigan)
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