Hasse Principles in algebraic equations
Heights of roots of polynomials with odd coefficients
Let be a zero of a polynomial of degree with odd coefficients, with not a root of unity. We show that the height of satisfiesMore generally, we obtain bounds when the coefficients are all congruent to modulo for some .
Heights of squares of Littlewood polynomials and infinite series
Let P be a unimodular polynomial of degree d-1. Then the height H(P²) of its square is at least √(d/2) and the product L(P²)H(P²), where L denotes the length of a polynomial, is at least d². We show that for any ε > 0 and any d ≥ d(ε) there exists a polynomial P with ±1 coefficients of degree d-1 such that H(P²) < (2+ε)√(dlogd) and L(P²)H(P²)< (16/3+ε)d²log d. A similar result is obtained for the series with ±1 coefficients. Let be the mth coefficient of the square f(x)² of a unimodular...
Higher Mahler measure of an n-variable family
We prove formulas for the k-higher Mahler measure of a family of rational functions with an arbitrary number of variables. Our formulas reveal relations with multiple polylogarithms evaluated at certain roots of unity.
Higher Mahler measures and zeta functions
Higher Newton polygons in the computation of discriminants and prime ideal decomposition in number fields
We present an algorithm for computing discriminants and prime ideal decomposition in number fields. The algorithm is a refinement of a -adic factorization method based on Newton polygons of higher order. The running-time and memory requirements of the algorithm appear to be very good.