Sparse sum-of-squares certificates on finite abelian groups

Let G be a finite abelian group. This paper is concerned with nonnegative functions on G that are sparse with respect to the Fourier basis. We establish combinatorial conditions on subsets S and T of Fourier basis elements under which nonnegative functions with Fourier support S are sums of squares of functions with Fourier support T. Our combinatorial condition involves constructing a chordal cover of a graph related to G and S (the Cayley graph Cay($\hat{G}$,S)) with maximal cliques related to T. Our result relies on two main ingredients: the decomposition of sparse positive semidefinite matrices with a chordal sparsity pattern, as well as a simple but key observation exploiting the structure of the Fourier basis elements of G. We apply our general result to two examples. First, in the case where $G = \mathbb{Z}_2^n$, by constructing a particular chordal cover of the half-cube graph, we prove that any nonnegative quadratic form in n binary variables is a sum of squares of functions of degree at most $\lceil n/2 \rceil$, establishing a conjecture of Laurent. Second, we consider nonnegative functions of degree d on $\mathbb{Z}_N$ (when d divides N). By constructing a particular chordal cover of the d'th power of the N-cycle, we prove that any such function is a sum of squares of functions with at most $3d\log(N/d)$ nonzero Fourier coefficients. Dually this shows that a certain cyclic polytope in $\mathbb{R}^{2d}$ with N vertices can be expressed as a projection of a section of the cone of psd matrices of size $3d\log(N/d)$. Putting $N=d^2$ gives a family of polytopes $P_d \subset \mathbb{R}^{2d}$ with LP extension complexity $\text{xc}_{LP}(P_d) = \Omega(d^2)$ and SDP extension complexity $\text{xc}_{PSD}(P_d) = O(d\log(d))$. To the best of our knowledge, this is the first explicit family of polytopes in increasing dimensions where $\text{xc}_{PSD}(P_d) = o(\text{xc}_{LP}(P_d))$.Comment: 34 page

Character sums with division polynomials

We obtain nontrivial estimates of quadratic character sums of division polynomials $\Psi_n(P)$, $n=1,2, ...$, evaluated at a given point $P$ on an elliptic curve over a finite field of $q$ elements. Our bounds are nontrivial if the order of $P$ is at least $q^{1/2 + \epsilon}$ for some fixed $\epsilon > 0$. This work is motivated by an open question about statistical indistinguishability of some cryptographically relevant sequences which has recently been brought up by K. Lauter and the second author

Iterative character constructions for algebra groups

We construct a family of orthogonal characters of an algebra group which decompose the supercharacters defined by Diaconis and Isaacs. Like supercharacters, these characters are given by nonnegative integer linear combinations of Kirillov functions and are induced from linear supercharacters of certain algebra subgroups. We derive a formula for these characters and give a condition for their irreducibility; generalizing a theorem of Otto, we also show that each such character has the same number of Kirillov functions and irreducible characters as constituents. In proving these results, we observe as an application how a recent computation by Evseev implies that every irreducible character of the unitriangular group \UT_n(q) of unipotent $n\times n$ upper triangular matrices over a finite field with $q$ elements is a Kirillov function if and only if $n\leq 12$. As a further application, we discuss some more general conditions showing that Kirillov functions are characters, and describe some results related to counting the irreducible constituents of supercharacters.Comment: 22 page

Generalising the Hardy-Littlewood Method for Primes

The Hardy-Littlewood method is a well-known technique in analytic number theory. Among its spectacular applications are Vinogradov's 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving an asymptotic for the number of 3-term progressions of primes, all less than N. This article surveys recent developments of the author and T. Tao, in which the Hardy-Littlewood method has been generalised to obtain, for example, an asymptotic for the number of 4-term arithmetic progressions of primes less than N.Comment: 26 pages, submitted to Proceedings of ICM 200