Factoring bivariate sparse (lacunary) polynomials

We present a deterministic algorithm for computing all irreducible factors of degree $\le d$ of a given bivariate polynomial $f\in K[x,y]$ over an algebraic number field $K$ and their multiplicities, whose running time is polynomial in the bit length of the sparse encoding of the input and in $d$. Moreover, we show that the factors over \Qbarra of degree $\le d$ which are not binomials can also be computed in time polynomial in the sparse length of the input and in $d$.Comment: 20 pp, Latex 2e. We learned on January 23th, 2006, that a multivariate version of Theorem 1 had independently been achieved by Erich Kaltofen and Pascal Koira

The number of maximal torsion cosets in subvarieties of tori

We present sharp bounds on the number of maximal torsion cosets in a subvariety of the complex algebraic torus $\mathbb{G}_{\textrm{m}}^n$. Our first main result gives a bound in terms of the degree of the defining polynomials. A second result gives a bound in terms of the toric degree of the subvariety. As a consequence, we prove the conjectures of Ruppert and of Aliev and Smyth on the number of isolated torsion points of a hypersurface. These conjectures bound this number in terms of the multidegree and the volume of the Newton polytope of a polynomial defining the hypersurface, respectively.Comment: 21 page

Finite Fields: Theory and Applications

Finite fields are the focal point of many interesting geometric, algorithmic and combinatorial problems. The workshop was devoted to progress on these questions, with an eye also on the important applications of finite field techniques in cryptography, error correcting codes, and random number generation

Measured Group Theory

The workshop aimed to study discrete and Lie groups and their actions using measure theoretic methods and their asymptotic invariants, such as $\ell^2$-invariants, the rank gradient, cost, torsion growth, entropy-type invariants and invariants coming from random walks and percolation theory. The participants came from a wide range of mathematics: asymptotic group theory, geometric group theory, ergodic theory, $\ell^2$-theory, graph convergence, representation theory, probability theory, descriptive set theory and algebraic topology

Factorization of polynomials in hyperbolic geometry and dynamics

Using factorization theorems for sparse polynomials, we compute the trace field of Dehn fillings of the Whitehead link, and (assuming Lehmer's Conjecture) the minimal polynomial of the small dilatation pseudo-Anosov maps and the trace field of fillings of the figure-8 knot. These results depend on the degrees of the trace fields over Q being sufficiently large.Comment: 16 pages, 1 figur

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