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Minimal primes over permanental ideals

George A. Kirkup

2008
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Transactions of the American Mathematical Society
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In this paper we discuss minimal primes over permanental ideals of generic matrices. We give a complete list of the minimal primes over ideals of 3×3 permanents of a generic matrix, and show that there are monomials in the ideal of maximal permanents of a d × (2d − 1) matrix if the characteristic of the ground field is sufficiently large. We also discuss the Alon-Jaeger-Tarsi Conjecture, using our results and techniques to strengthen the previously known results. License or copyright
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... opyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 3752 GEORGE A. KIRKUP 1.3. Overview. We are motivated by both [LS] and [AT]. Our techniques will be very algebraic, and will be simple applications of the multilinearity of permanents which is discussed in Section 2. In Section 3 we review relevant results from [LS] and give several proofs based on the results of the previous section. In Section 4 we discuss the main conjecture of this paper, which is Conjecture 1. If char k > d or char k = 0 and n ≥ 2d − 1, then the minimal primes containing I d (m, n) must either contain a column of the generic matrix or This would inductively give the minimal primes over the ideals I d (m, n) for large m, n if we knew the minimal primes over I d (m, n) for m, n ≤ 2d − 1. We prove Conjecture 1 for the case d = 3 in Section 5. What we are able to show in general is that any prime over I d (d, 2d − 1) which contains no entry from some row must contain the (d − 1)×(d − 1) permanents of the other d − 1 rows. We continue in Section 6 where we discuss the ideals I d (d, d + 1) in detail. Then in Section 7 we focus on I 3 (3, 4), paying special attention to 3×4 matrices with no entries vanishing whose maximal permanents do vanish. In Section 8 we discuss in general the case in which m, n > d. We apply these ideas to the case d = 3 in Section 9, in which we list the minimal primes over I 3 (4, 4). From this we deduce the minimal primes over I 3 (m, n) for all m, n. This classification is the main result of our general study of permanental ideals. By way of introduction, we give a very weak form of Theorem 16. Corollary 2. Suppose A is an m×n matrix, over a field of characteristic other than 2 or 3, with m, n ≥ 4 whose 3×3 subpermanents vanish. Either a row of A is identically 0, a column of A is identically 0, there is an (m − 1)×(n − 1) submatrix of A which is identically 0, or A is a 4×4 matrix with two disjoint 2×2 blocks that are identically 0. In Section 10 we relate our conjecture and results to the combinatorial conjectures in [AT]. The main conjecture of [AT] is Conjecture 3. Let A be a nonsingular d×d matrix over a finite field k with cardinality q ≥ 4. There exists a vector v in k n such that both v and Av have no zero component. The connection between permanents and this conjecture was made in [AT], and we will present that connection in detail. After this we apply our techniques to strengthen the results of [AT] and [BBLS]. The multilinearity of the permanent and the algebra of permanents Let A = (a i,j ) be a d×d matrix. Then the permanent of M can be expressed as d 1 x i,1 · Aî1

doi:10.1090/s0002-9947-08-04340-7
fatcat:2mu5dpq2kvdarl6nau27qdsjqm