A Study on a New Method of Dynamic Aperture Enlargement

TL;DR

This paper proposes a novel method for enlarging the dynamic aperture of particle accelerators by approximating generating functions to move all resonances outward simultaneously, involving stable polynomials and determinantal representations.

Contribution

It introduces a new approach to dynamic aperture enlargement that moves all resonances concurrently using polynomial approximation and advances understanding of stable polynomial representations.

Findings

Progress towards constructing symmetric determinantal representations of multivariable polynomials.

Development of methods to approximate generating functions to control resonance locations.

Exploration of algebraic tools like Gr"obner bases for analyzing polynomial gradients.

Abstract

This report summarizes progress made towards a new approach for enlarging the dynamic aperture of particle accelerators. Unlike prior methods which attempted to move the location of select resonances outward in phase space, our approach aims to move all resonances concurrently. These resonances are in one-to-one correspondence with fixed points of symplectic maps, which in turn are in a one-to-one correspondence with the critical points of their generating function. Thus in this approach, the problem of enlarging dynamic aperture boils down to an approximation problem: given a generating function, approximate it by a function whose critical points are outside a specified elliptical region. In attempting to solve the generating function approximation problem, we employed stable polynomials. Many stable polynomials have a determinantal representation that indicates stability. However, it is an open question as to whether such a determinantal representation can be found for a given stable polynomial. In seeking to answer this, we made progress towards constructing a symmetric determinantal representations of multivariable polynomials with the smallest sized linear pencil. This report also contains brief surveys of topics including Clifford numbers, stable polynomials, and symmetric determinantal representations of polynomials. We also explored using Gr\"obner bases to find where the gradient of a multivariable polynomial is zero.