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Register a new userDensity of real and complex decomposable univariate polynomials
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We estimate the density of tubes around the algebraic variety of decomposable univariate polynomials over the real and the complex numbers.
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The usual univariate interpolation problem of finding a monic polynomial f of degree n that interpolates n given values is well understood. This paper studies a variant where f is required to be composite, say, a composition of two polynomials of degrees d and e, respectively, with de=n, and therefore d+e-1 given values. Some special cases are easy to solve, and for the general case, we construct a homotopy between it and a special case. We compute a geometric solution of the algebraic curve presenting this homotopy, and this also provides an answer to the interpolation task. The computing time is polynomial in the geometric data, like the degree, of this curve. A consequence is that for almost all inputs, a decomposable interpolation polynomial exists.
A univariate trace polynomial is a polynomial in a variable x and formal trace symbols Tr(x^j). Such an expression can be naturally evaluated on matrices, where the trace symbols are evaluated as normalized traces. This paper addresses global and constrained positivity of univariate trace polynomials on symmetric matrices of all finite sizes. A tracial analog of Artins solution to Hilberts 17th problem is given: a positive semidefinite univariate trace polynomial is a quotient of sums of products of squares and traces of squares of trace polynomials.
We calculate the E-polynomial for a class of the (complex) character varieties $mathcal{M}_n^{tau}$ associated to a genus $g$ Riemann surface $Sigma$ equipped with an orientation reversing involution $tau$. Our formula expresses the generating function $sum_{n=1}^{infty} E(mathcal{M}_n^{tau}) T^n$ as the plethystic logarithm of a product of sums indexed by Young diagrams. The proof uses point counting over finite fields, emulating Hausel and Rodriguez-Villegas.
We initiate the study of random iteration of automorphisms of real and complex projective surfaces, or more generally compact K{a}hler surfaces, focusing on the fundamental problem of classification of stationary measures. We show that, in a number of cases, such stationary measures are invariant, and provide criteria for uniqueness, smoothness and rigidity of invariant probability measures. This involves a variety of tools from complex and algebraic geometry, random products of matrices, non-uniform hyperbolicity, as well as recent results of Brown and Rodriguez Hertz on random iteration of surface diffeomorphisms.
Amendola et al. proposed a method for solving systems of polynomial equations lying in a family which exploits a recursive decomposition into smaller systems. A family of systems admits such a decomposition if and only if the corresponding Galois group is imprimitive. When the Galois group is imprimitive we consider the problem of computing an explicit decomposition. A consequence of Esterovs classification of sparse polynomial systems with imprimitive Galois groups is that this decomposition is obtained by inspection. This leads to a recursive algorithm to solve decomposable sparse systems, which we present and give evidence for its efficiency.
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