Generic polynomial

About: Generic polynomial is a research topic. Over the lifetime, 608 publications have been published within this topic receiving 6784 citations.

Root numbers, Selmer groups, and non-commutative Iwasawa theory

Abstract: Let E be an elliptic curve over a number field F, and let F ∞ be a Galois extension of F whose Galois group G is a p-adic Lie group. The aim of the present paper is to provide some evidence that, in accordance with the main conjectures of Iwasawa theory, there is a close connection between the action of the Selmer group of E over F ∞ , and the global root numbers attached to the twists of the complex L-function of E by Artin representations of G.

Algebraic Bethe ansatz for a discrete-state BCS pairing model

Abstract: We show in detail how Richardson’s exact solution of a discrete-state BCS ~DBCS! model can be recovered as a special case of an algebraic Bethe-ansatz solution of the inhomogeneous XXX vertex model with twisted boundary conditions: by implementing the twist using Sklyanin’s K-matrix construction and taking the quasiclassical limit, one obtains a complete set of conserved quantities Hi from which the DBCS Hamiltonian can be constructed as a second order polynomial. The eigenvalues and eigenstates of the Hi ~which reduce to the Gaudin Hamiltonians in the limit of infinitely strong coupling! are exactly known in terms of a set of parameters determined by a set of on-shell Bethe ansatz equations, which reproduce Richardson’s equations for these parameters. We thus clarify that the integrability of the DBCS model is a special case of the integrability of the twisted inhomogeneous XXX vertex model. Furthermore, by considering the twisted inhomogeneous XXZ model and/or choosing a generic polynomial of the Hi’s as Hamiltonian, more general exactly solvable models can be constructed. To make the paper accessible to readers that are not Bethe-ansatz experts, the introductory sections include a self-contained review of those of its feature which are needed here.

Functoriality and the inverse Galois problem

Abstract: We prove that, for any primeand any even integer n, there are infinitely many exponents k for which PSp n (Fk) appears as a Galois group over Q. This generalizes a result of Wiese from 2006, which inspired this paper.

Sequential Galois multiplication in GF(2n) with GF(2m) Galois multiplication gates

Abstract: Configurations of Boolean elements for implementing a sequential GF(2n) Galois multiplication gate are disclosed. Each configuration includes a single subfield GF(2m) Galois multiplication gate, where m is a positive integral divisor of n, e.g., n=8 and m=2, and assorted controls. Also disclosed is a sequential implementation of a GF(2n) Galois linear module as described in the J. T. Ellison Pat. No. 3,805,037 wherein the controls of the sequential GF(2n) multiply gate cause the Galois addition (bit-wise Exclusive-OR) of an n-bit binary vector, Z, to the final Galois product.

Polynomial phase masks for extending the depth of field of a microscope.

Abstract: A polynomial phase mask is designed and fabricated for enhancing the depth of field of a microscope by more than tenfold. A generic polynomial of degree 31 that consists of 16 odd power terms is optimized by simulated annealing with a realistic average modulation transfer function (MTF) iteratively set as the target MTF. Optical experimental results are shown.