Length of polynomials over finite groups

This document is the Accepted Manuscript version of the following article: Gábor Horváth, and Chrystopher L. Nehaniv, ‘Length of polynomials over finite groups’, Journal of Computer and System Sicences, Vol. 81(8): 1614-1622, December 2015. The final, published version is available online at doi: https://doi.org/10.1016/j.jcss.2015.05.002.We study the length of polynomials over finite simple non-Abelian groups needed to realize Boolean functions. We apply the results for bounding the length of 5-permutation branching programs recognizing a Boolean set. Moreover, for Boolean and general functions on these groups, we present upper bounds on the length of shortest polynomials computing an arbitrary n-ary Boolean or general function, or a function given by another polynomial.Peer reviewe

Minimal length elements of extended affine Coxeter groups, II

Let $W$ be an extended affine Weyl group. We prove that minimal length elements w_{\co} of any conjugacy class \co of $W$ satisfy some special properties, generalizing results of Geck and Pfeiffer \cite{GP} on finite Weyl groups. We then introduce the "class polynomials" for affine Hecke algebra $H$ and prove that T_{w_\co}, where \co runs over all the conjugacy classes of $W$, forms a basis of the cocenter $H/[H, H]$. We also classify the conjugacy classes satisfying a generalization of Lusztig's conjecture \cite{L4}.Comment: 32 page

Finite groups acting linearly: Hochschild cohomology and the cup product

When a finite group acts linearly on a complex vector space, the natural semi-direct product of the group and the polynomial ring over the space forms a skew group algebra. This algebra plays the role of the coordinate ring of the resulting orbifold and serves as a substitute for the ring of invariant polynomials from the viewpoint of geometry and physics. Its Hochschild cohomology predicts various Hecke algebras and deformations of the orbifold. In this article, we investigate the ring structure of the Hochschild cohomology of the skew group algebra. We show that the cup product coincides with a natural smash product, transferring the cohomology of a group action into a group action on cohomology. We express the algebraic structure of Hochschild cohomology in terms of a partial order on the group (modulo the kernel of the action). This partial order arises after assigning to each group element the codimension of its fixed point space. We describe the algebraic structure for Coxeter groups, where this partial order is given by the reflection length function; a similar combinatorial description holds for an infinite family of complex reflection groups.Comment: 30 page

Finitely ramified iterated extensions

Let K be a number field, t a parameter, F=K(t) and f in K[x] a polynomial of degree d. The polynomial P_n(x,t)= f^n(x) - t in F[x] where f^n is the n-fold iterate of f, is absolutely irreducible over F; we compute a recursion for its discriminant. Let L=L(f) be the field obtained by adjoining to F all roots, in a fixed algebraic closure, of P_n for all n; its Galois group Gal(L/F) is the iterated monodromy group of f. The iterated extension L/F is finitely ramified if and only if f is post-critically finite (pcf). We show that, moreover, for pcf polynomials f, every specialization of L/F at t=t_0 in K is finitely ramified over K, pointing to the possibility of studying Galois groups with restricted ramification via tree representations associated to iterated monodromy groups of pcf polynomials. We discuss the wildness of ramification in some of these representations, describe prime decomposition in terms of certain finite graphs, and also give some examples of monogene number fields.Comment: 19 page

The set of stable primes for polynomial sequences with large Galois group

Let $K$ be a number field with ring of integers $\mathcal O_K$, and let $\{f_k\}_{k\in \mathbb N}\subseteq \mathcal O_K[x]$ be a sequence of monic polynomials such that for every $n\in \mathbb N$, the composition $f^{(n)}=f_1\circ f_2\circ\ldots\circ f_n$ is irreducible. In this paper we show that if the size of the Galois group of $f^{(n)}$ is large enough (in a precise sense) as a function of $n$, then the set of primes $\mathfrak p\subseteq\mathcal O_K$ such that every $f^{(n)}$ is irreducible modulo $\mathfrak p$ has density zero. Moreover, we prove that the subset of polynomial sequences such that the Galois group of $f^{(n)}$ is large enough has density 1, in an appropriate sense, within the set of all polynomial sequences.Comment: Comments are welcome