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Factoring bivariate lacunary polynomials without heights
, 2013
"... 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 P (X) =∑k j=1 ajX αj (1+X)βj is identically zero in polynomial time. The algorithm we obtain is more elementary than the one by Kaltofen and ..."
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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 P (X) =∑k j=1 ajX αj (1+X)βj is identically zero in polynomial time. 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.
FEWNOMIAL SYSTEMS WITH MANY ROOTS, AND AN ADELIC TAU CONJECTURE
"... Abstract. Consider a system F of n polynomials in 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 nondegenerate roots F can have over L, with all coordinates having fixed phase, as a function of n, k, ..."
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Abstract. Consider a system F of n polynomials in 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 nondegenerate 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 briefly review the background behind such bounds, and their application, including connections to computational number theory and variants of the ShubSmale τConjecture and the P vs. NP Problem. One of our key tools is the construction of combinatorially constrained tropical varieties with maximally many intersections.
Boundeddegree factors of lacunary multivariate polynomials
, 2014
"... Abstract. In this paper, we present a new method for computing boundeddegree factors of lacunary multivariate polynomials. In particular for polynomials over number fields, we give a new algorithm that takes as input a multivariate polynomial f in lacunary representation and a degree bound d and co ..."
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Abstract. In this paper, we present a new method for computing boundeddegree factors of lacunary multivariate polynomials. In particular for polynomials over number fields, we give a new algorithm that takes as input a multivariate polynomial f in lacunary representation and a degree bound d and computes the irreducible factors of degree at most d of f in time polynomial in the lacunary size of f and in d. Our algorithm consists in a reduction of the problem to the univariate case on the one hand and to the irreducible factorization of multivariate lowdegree polynomials on the other hand, which is valid for any field of zero characteristic. The reduction algorithms we propose are elementary in that they only manipulate the exponent vectors of the input polynomial. The proof of correctness and the complexity bounds rely on the valuations of certain formal power series with rational exponents, called Puiseux series, and on considerations on the Newton polytope of the polynomial. 1.