A sum-product theorem in function fields

Let $A$ be a finite subset of \ffield, the field of Laurent series in $1/t$ over a finite field $\mathbb{F}_q$. We show that for any $\epsilon>0$ there exists a constant $C$ dependent only on $\epsilon$ and $q$ such that $\max\{|A+A|,|AA|\}\geq C |A|^{6/5-\epsilon}$. In particular such a result is obtained for the rational function field $\mathbb{F}_q(t)$. Identical results are also obtained for finite subsets of the $p$-adic field $\mathbb{Q}_p$ for any prime $p$.Comment: Simplification of argument and note that methods also work for the p-adic

Tropical Fourier-Motzkin elimination, with an application to real-time verification

We introduce a generalization of tropical polyhedra able to express both strict and non-strict inequalities. Such inequalities are handled by means of a semiring of germs (encoding infinitesimal perturbations). We develop a tropical analogue of Fourier-Motzkin elimination from which we derive geometrical properties of these polyhedra. In particular, we show that they coincide with the tropically convex union of (non-necessarily closed) cells that are convex both classically and tropically. We also prove that the redundant inequalities produced when performing successive elimination steps can be dynamically deleted by reduction to mean payoff game problems. As a complement, we provide a coarser (polynomial time) deletion procedure which is enough to arrive at a simply exponential bound for the total execution time. These algorithms are illustrated by an application to real-time systems (reachability analysis of timed automata).Comment: 29 pages, 8 figure

Tropical polyhedra are equivalent to mean payoff games

We show that several decision problems originating from max-plus or tropical convexity are equivalent to zero-sum two player game problems. In particular, we set up an equivalence between the external representation of tropical convex sets and zero-sum stochastic games, in which tropical polyhedra correspond to deterministic games with finite action spaces. Then, we show that the winning initial positions can be determined from the associated tropical polyhedron. We obtain as a corollary a game theoretical proof of the fact that the tropical rank of a matrix, defined as the maximal size of a submatrix for which the optimal assignment problem has a unique solution, coincides with the maximal number of rows (or columns) of the matrix which are linearly independent in the tropical sense. Our proofs rely on techniques from non-linear Perron-Frobenius theory.Comment: 28 pages, 5 figures; v2: updated references, added background materials and illustrations; v3: minor improvements, references update

Variations on the Sum-Product Problem

This paper considers various formulations of the sum-product problem. It is shown that, for a finite set $A\subset{\mathbb{R}}$, $$|A(A+A)|\gg{|A|^{\frac{3}{2}+\frac{1}{178}}},$$ giving a partial answer to a conjecture of Balog. In a similar spirit, it is established that $$|A(A+A+A+A)|\gg{\frac{|A|^2}{\log{|A|}}},$$ a bound which is optimal up to constant and logarithmic factors. We also prove several new results concerning sum-product estimates and expanders, for example, showing that $$|A(A+a)|\gg{|A|^{3/2}}$$ holds for a typical element of $A$.Comment: 30 pages, new version contains improved exponent in main theorem due to suggestion of M. Z. Garae

The level set method for the two-sided eigenproblem

We consider the max-plus analogue of the eigenproblem for matrix pencils Ax=lambda Bx. We show that the spectrum of (A,B) (i.e., the set of possible values of lambda), which is a finite union of intervals, can be computed in pseudo-polynomial number of operations, by a (pseudo-polynomial) number of calls to an oracle that computes the value of a mean payoff game. The proof relies on the introduction of a spectral function, which we interpret in terms of the least Chebyshev distance between Ax and lambda Bx. The spectrum is obtained as the zero level set of this function.Comment: 34 pages, 4 figures. Changes with respect to the previous version: we explain relation to mean-payoff games and discrete event systems, and show that the reconstruction of spectrum is pseudopolynomia

Zeros of polynomials over finite Witt rings

Let $\mathbb{F}_q$ denote the finite field of characteristic $p$ and order $q$. Let $\mathbb{Z}_q$ denote the unramified extension of the $p$-adic rational integers $\mathbb{Z}_p$ with residue field $\mathbb{F}_q$. Given two positive integers $m,n$, define a box $\mathcal B_m$ to be a subset of $\mathbb{Z}_q^n$ with $q^{nm}$ elements such that $\mathcal B_m$ modulo $p^m$ is equal to $(\mathbb{Z}_q/p^m \mathbb{Z}_q)^n$. For a collection of nonconstant polynomials $f_1,\dots,f_s\in \mathbb{Z}_q[x_1,\ldots,x_n]$ and positive integers $m_1,\dots,m_s$, define the set of common zeros inside the box $\mathcal B_m$ to be V=\{X\in \mathcal B_m:\; f_i(X)\equiv 0\mod {p^{m_i}}\mbox{ for all } 1\leq i\leq s\}. It is an interesting problem to give the sharp estimates for the $p$-divisibility of $|V|$. This problem has been partially solved for the three cases: (i) $m=m_1=\cdots=m_s=1$, which is just the Ax-Katz theorem, (ii) $m=m_1=\cdots=m_s>1$, which was solved by Katz, Marshal and Ramage, and (iii) $m=1$, and $m_1,\dots,m_s\geq 1$, which was recently solved by Cao, Wan and Grynkiewicz. Based on the multi-fold addition and multiplication of the finite Witt rings over $\mathbb{F}_q$, we investigate the remaining unconsidered case of $m>1$ and $m\neq m_j$ for some $1\leq j\leq s$, and finally provide a complete answer to this problem