157 research outputs found
Computing low-degree factors of lacunary polynomials: a Newton-Puiseux approach
We present a new algorithm for the computation of the irreducible factors of
degree at most , with multiplicity, of multivariate lacunary polynomials
over fields of characteristic zero. The algorithm reduces this computation to
the computation of irreducible factors of degree at most of univariate
lacunary polynomials and to the factorization of low-degree multivariate
polynomials. The reduction runs in time polynomial in the size of the input
polynomial and in . As a result, we obtain a new polynomial-time algorithm
for the computation of low-degree factors, with multiplicity, of multivariate
lacunary polynomials over number fields, but our method also gives partial
results for other fields, such as the fields of -adic numbers or for
absolute or approximate factorization for instance.
The core of our reduction uses the Newton polygon of the input polynomial,
and its validity is based on the Newton-Puiseux expansion of roots of bivariate
polynomials. In particular, we bound the valuation of where is
a lacunary polynomial and a Puiseux series whose vanishing polynomial
has low degree.Comment: 22 page
A hitting set construction, with application to arithmetic circuit lower bounds
14 pagesA polynomial identity testing algorithm must determine whether a given input polynomial is identically equal to 0. We give a deterministic black-box identity testing algorithm for univariate polynomials of the form . From our algorithm we derive an exponential lower bound for representations of polynomials such as under this form. It has been conjectured that these polynomials are hard to compute by general arithmetic circuits. Our result shows that the ``hardness from derandomization'' approach to lower bounds is feasible for a restricted class of arithmetic circuits. The proof is based on techniques from algebraic number theory, and more precisely on properties of the height function of algebraic numbers
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