Factoring bivariate lacunary polynomials without heights

We present an algorithm which computes the multilinear factors of bivariate lacunary polynomials. It is based on a new Gap Theorem which allows to test whether a polynomial of the form P(X,X+1) is identically zero in time polynomial in the number of terms of P(X,Y). The algorithm we obtain is more elementary than the one by Kaltofen and Koiran (ISSAC'05) since it relies on the valuation of polynomials of the previous form instead of the height of the coefficients. As a result, it can be used to find some linear factors of bivariate lacunary polynomials over a field of large finite characteristic in probabilistic polynomial time.Comment: 25 pages, 1 appendi

How many zeros of a random sparse polynomial are real?

We investigate the number of real zeros of a univariate $k$-sparse polynomial $f$ over the reals, when the coefficients of $f$ come from independent standard normal distributions. Recently B\"urgisser, Erg\"ur and Tonelli-Cueto showed that the expected number of real zeros of $f$ in such cases is bounded by $O(\sqrt{k} \log k)$. In this work, we improve the bound to $O(\sqrt{k})$ and also show that this bound is tight by constructing a family of sparse support whose expected number of real zeros is lower bounded by $\Omega(\sqrt{k})$. Our main technique is an alternative formulation of the Kac integral by Edelman-Kostlan which allows us to bound the expected number of zeros of $f$ in terms of the expected number of zeros of polynomials of lower sparsity. Using our technique, we also recover the $O(\log n)$ bound on the expected number of real zeros of a dense polynomial of degree $n$ with coefficients coming from independent standard normal distributions

Quantitative Aspects of Sums of Squares and Sparse Polynomial Systems

Computational algebraic geometry is the study of roots of polynomials and polynomial systems. We are familiar with the notion of degree, but there are other ways to consider a polynomial: How many variables does it have? How many terms does it have? Considering the sparsity of a polynomial means we pay special attention to the number of terms. One can sometimes profit greatly by making use of sparsity when doing computations by utilizing tools from linear programming and integer matrix factorization. This thesis investigates several problems from the point of view of sparsity. Consider a system F of n polynomials over n variables, with a total of n + k distinct exponent vectors over any local field L. We discuss conjecturally tight bounds on the maximal number of non-degenerate roots F can have over L, with all coordinates having fixed phase, as a function of n, k, and L only. In particular, we give new explicit systems with number of roots approaching the best known upper bounds. We also give a complete classification for when an n-variate n + 2-nomial positive polynomial can be written as a sum of squares of polynomials. Finally, we investigate the problem of approximating roots of polynomials from the viewpoint of sparsity by developing a method of approximating roots for binomial systems that runs more efficiently than other current methods. These results serve as building blocks for proving results for less sparse polynomial systems

A Wronskian approach to the real τ-conjecture

International audienc