Linear Equations with Rational Fractions of Bounded Height and Stochastic Matrices

TL;DR

This paper establishes near-optimal bounds on the number of solutions to certain linear equations involving rational fractions and applies these results to estimate the quantity of stochastic matrices with rational entries of bounded height.

Contribution

It provides a tight upper bound on solutions to rational fraction equations and applies this to count stochastic matrices with bounded rational entries.

Findings

Bound on solutions is tight up to a logarithmic factor.

Derived an upper bound on the number of stochastic matrices with bounded rational entries.

Applicable to equations with variables in arbitrary boxes and translations.

Abstract

We obtain a tight, up to a logarithmic factor, upper bound on the number of solutions to the equation $$ \sum_{j=1}^n a_j \frac{s_j}{r_j} =a_0, \qquad $$ with variables $r_1,...,r_n$ in an arbitrary box at the origin and variables $s_1,..., s_n$ in an essentially arbitrary translation of this box. We apply this result to get an upper bound on the number of stochastic matrices with rational entries of bounded height.