In this paper we study how the number of integer points in a polytope grows as we dilate the polytope. We prove new and essentially tight bounds on this quantity by specifically studying dilates of the Hadamard polytope. Our motivation for studying this quantity comes from the problem of understanding the maximal number of monomials in a factor of a multivariate polynomial with $s$ monomials. A recent result by Bhargava, Saraf, and Volkovich showed that if $f$ is an $n$-variate polynomial, where each variable has degree $d$, and $f$ has $s$ monomials, then any factor of $f$ has at most $s^{O(d^2 \log n)}$ monomials. The key technical ingredient of their proof was to show that any polytope with $s$ vertices, where each vertex lies in $\{0,..,d\}^n$, can have at most $s^{O(d^2 \log n)}$ integer points. The precise dependence on $d$ of the number of integer points was left open. We show that this bound, particularly the dependence on $d$, is essentially tight by studying dilates of the Hadamard polytope and proving new lower bounds on the number of its integer points.

In this paper, we give the first subexponential (in fact, quasi-polynomial time) reconstruction algorithm for depth-3 circuits of any constant top fan-in (ΣΠΣ(k) circuits) over ℝ, ℂ, or any large characteristic finite field F. More explicitly, we show that for any constant k, given black-box access to an n-variate polynomial f computed by a ΣΠΣ(k) circuit of size s, there is a randomized algorithm that runs in time quasi-poly(n,s) and outputs a generalized ΣΠΣ(k) circuit computing f. The size s includes the bit complexity of coefficients appearing in the circuit: this is the max bit complexity if the field is ℝ or ℂ, and log|F| if the field is finite.