Christopher J. Hillar

Automorphisms of Finite Abelian Groups

Much less known, however, is that there is a description of Aut(G), the automorphism group of G. The first compete characterization that we are aware of is contained in a paper by Ranum [1] near the turn of the last century. Beyond this, however, there are few other expositions [4]. Our goal is to fill this gap, thereby providing a much needed accessible and modern treatment. Our characterization of Aut(G) is accomplished in three main steps. The first observation is that it is enough to work with the simpler groups Hp. This reduction is carried out by appealing to a fact about product automorphisms for groups with relatively prime numbers of elements (Lemma 2.1). Next, we use Theorem 3.3 to describe the endomorphism ring of Hp as a quotient of a matrix subring of Zn×n. And finally, the units Aut(Hp) ⊂ End(Hp) are identified from this construction. As a consequence of our investigation, we readily obtain an explicit formula for the number of elements of Aut(G) for any finite Abelian group G (see also [3]).

Finite generation of symmetric ideals

Let be a commutative Noetherian ring, and let be the polynomial ring in an infinite collection of indeterminates over . Let be the group of permutations of . The group acts on in a natural way, and this in turn gives the structure of a left module over the group ring . We prove that all ideals of invariant under the action of are finitely generated as -modules. The proof involves introducing a certain well-quasi-ordering on monomials and developing a theory of Grobner bases and reduction in this setting. We also consider the concept of an invariant chain of ideals for finite-dimensional polynomial rings and relate it to the finite generation result mentioned above. Finally, a motivating question from chemistry is presented, with the above framework providing a suitable context in which to study it.

Algebraic characterization of uniquely vertex colorable graphs

The study of graph vertex colorability from an algebraic perspective has introduced novel techniques and algorithms into the field. For instance, it is known that k-colorability of a graph G is equivalent to the condition 1@?IG,k for a certain ideal IG,k@?k[x1,...,xn]. In this paper, we extend this result by proving a general decomposition theorem for IG,k. This theorem allows us to give an algebraic characterization of uniquely k-colorable graphs. Our results also give algorithms for testing unique colorability. As an application, we verify a counterexample to a conjecture of Xu concerning uniquely 3-colorable graphs without triangles.

AN ELEMENTARY AND CONSTRUCTIVE SOLUTION TO HILBERT'S 17TH PROBLEM FOR MATRICES

We give a short and elementary proof of a theorem of Procesi, Schacher and (independently) Gondard, Ribenboim that generalizes a famous result of Artin. Let A be an n x n symmetric matrix with entries in the polynomial ring R[x 1 ,..,x m ]. The result is that if A is positive semidefinite for all substitutions (x 1 ,...,x m ) ∈ R m , then A can be expressed as a sum of squares of symmetric matrices with entries in R(x 1 ,...,x m ). Moreover, our proof is constructive and gives explicit representations modulo the scalar case.

Polynomial recurrences and cyclic resultants

Let K be an algebraically closed field of characteristic zero and let f ∈ K[x]. The m-th cyclic resultant of f is r m = Res(f , x m - 1). A generic monic polynomial is determined by its full sequence of cyclic resultants; however, the known techniques proving this result give no effective computational bounds. We prove that a generic monic polynomial of degree d is determined by its first 2 d+1 cyclic resultants and that a generic monic reciprocal polynomial of even degree d is determined by its first 2 3 d/2 of them. In addition, we show that cyclic resultants satisfy a polynomial recurrence of length d+ 1. This result gives evidence supporting the conjecture of Sturmfels and Zworski that d + 1 resultants determine f. In the process, we establish two general results of independent interest: we show that certain Toeplitz determinants are sufficient to determine whether a sequence is linearly recurrent, and we give conditions under which a linearly recurrent sequence satisfies a polynomial recurrence of shorter length.

Sums of squares over totally real fields are rational sums of squares

Let K be a totally real number field with Galois closure L. We prove that if f ∈ Q(x 1 ,.., x n ] is a sum of m squares in K[x 1 ,.., x n ], then f is a sum of 4m·2 [L:Q]+1 ( [L:Q+1 2 ) squares in Q[x 1 ,... x n ]. Moreover, our argument is constructive and generalizes to the case of commutative K-algebras. This result gives a partial resolution to a question of Sturmfels on the algebraic degree of certain semi-definite programming problems.

An algorithm for finding symmetric Grobner bases in infinite dimensional rings

A symmetric ideal I ⊂ R = K[x1,x2,...] is an ideal that is invariant under the natural action of the infinite symmetric group. We give an explicit algorithm to find Grobner bases for symmetric ideals in the infinite dimensional polynomial ring R. This allows for symbolic computation in a new class of rings. In particular, we solve the ideal membership problem for symmetric ideals of R.

Minimal generators for symmetric ideals

Let R = K[X] be the polynomial ring in infinitely many indeterminates X over a field K, and let G X be the symmetric group of X. The group G X acts naturally on R, and this in turn gives R the structure of a module over the group ring R[G X ] A recent theorem of Aschenbrenner and Hillar states that the module R is Noetherian. We address whether submodules of R can have any number of minimal generators, answering this question positively.

Two Sums That Are Congruent Modulo p: 10723

Editorial comment. The three expressions in (1) are congruent to 0 mod p if and only if 2P-1 1 (mod p2). Such primes are known as Wieferich primes and are related to Fermats Last Theorem (A. Wieferich, J. Reine Angew. Math. 136 (1909) 293-302) and other number theory problems. R. E. Crandall, K. Dilcher, and C. Pomerance (Math. Comp. 66 (1997) 433-449) have verified by direct computations that the only Wieferich primes less than 4 x 1012 are 1093 and 3511. The GCHQ Problems Group mentioned that (1) follows from two results in L. E. Dicksons History of the Theory of Numbers, Chelsea, 1966. T. M. Putnam (this MONTHLY 21 (1914) 220-222) showed the first congruence in (1), while P. Bachman (J. Reine Angew. Math. 142 (1913) 41-50) showed that ESf1I 21/(p i) = 2P2/p (mod p), which implies the second congruence in (1). R. Mandl generalized the first congruence in (1) (the same proof works):