A ug 2 00 6 Search techniques for root - unitary polynomials
نویسنده
چکیده
We give an anecdotal discussion of the problem of searching for polynomials with all roots on the unit circle, whose coefficients are rational numbers subject to certain congruence conditions. We illustrate with an example from a calculation in p-adic cohomology made by Abbott, Kedlaya, and Roe, in which (using an implementation developed with Andre Wibisono) we recover the zeta function of a surface over a finite field. Introduction In this note, we give an anecdotal discussion of the problem of searching for polynomials with roots on a prescribed circle whose coefficients are rational numbers subject to certain congruence conditions. We were led to this problem by the use of p-adic cohomology to compute zeta functions of varieties over finite fields; in that context, one is looking for certain Weil polynomials (monic integer polynomials with complex roots all on a circle of radius p, for some prime number p and some nonnegative integer i), and the cohomology calculation imposes congruence conditions on the coefficients. In fact, the main purpose of this note is to show that in a particular example from [1], the conditions obtained from the cohomology calculation indeed suffice to uniquely determine the zeta function being sought. 1 Definitions A polynomial P (z) = ∑n i=0 aiz i ∈ C[z] of degree n is self-inversive if there exists u ∈ C with |u| = 1 such that ai = uan−i (i = 0, . . . , n); (1.1)
منابع مشابه
ar X iv : 0 90 8 . 01 53 v 1 [ m at h . G T ] 2 A ug 2 00 9 On Fibonacci knots
We show that the Conway polynomials of Fibonacci links are Fibonacci polynomials modulo 2. We deduce that, when n 6≡ 0 (mod 4) and (n, j) 6= (3, 3), the Fibonacci knot F (n) j is not a Lissajous knot. keywords: Fibonacci polynomials, Fibonacci knots, continued fractions
متن کاملA ug 2 00 3 SELF - SELF - DUAL SPACES OF POLYNOMIALS
A space of polynomials V of dimension 7 is called self-dual if the divided Wronskian of any 6-subspace is in V . A self-dual space V has a natural inner product. The divided Wronskian of any isotropic 3-subspace of V is a square of a polynomial. We call V self-self-dual if the square root of the divided Wronskian of any isotropic 3-subspace is again in V . We show that the self-self-dual spaces...
متن کاملSearch techniques for root-unitary polynomials
We give an anecdotal discussion of the problem of searching for polynomials with all roots on the unit circle, whose coefficients are rational numbers subject to certain congruence conditions. We illustrate with an example from a calculation in p-adic cohomology made by Abbott, Kedlaya, and Roe, in which we recover the zeta function of a surface over a finite field.
متن کاملar X iv : h ep - e x / 00 08 06 5 v 1 2 6 A ug 2 00 0 Search for Large Extra Dimensions in Dielectron and Diphoton Production
متن کامل
ar X iv : m at h - ph / 0 50 80 68 v 1 3 1 A ug 2 00 5 LAMÉ EQUATION , QUANTUM TOP AND ELLIPTIC BERNOULLI POLYNOMIALS
A generalisation of the odd Bernoulli polynomials related to the quantum Euler top is introduced and investigated. This is applied to compute the coefficients of the spectral polynomials for the classical Lamé operator.
متن کامل