Numerical Algebraic Geometry: A New Perspective on String and Gauge Theories

The interplay rich between algebraic geometry and string and gauge theories has recently been immensely aided by advances in computational algebra. However, these symbolic (Gr\"{o}bner) methods are severely limited by algorithmic issues such as exponential space complexity and being highly sequential. In this paper, we introduce a novel paradigm of numerical algebraic geometry which in a plethora of situations overcomes these short-comings. Its so-called 'embarrassing parallelizability' allows us to solve many problems and extract physical information which elude the symbolic methods. We describe the method and then use it to solve various problems arising from physics which could not be otherwise solved.Comment: 36 page

CHAMP: A Cherednik Algebra Magma Package

We present a computer algebra package based on Magma for performing computations in rational Cherednik algebras at arbitrary parameters and in Verma modules for restricted rational Cherednik algebras. Part of this package is a new general Las Vegas algorithm for computing the head and the constituents of a module with simple head in characteristic zero which we develop here theoretically. This algorithm is very successful when applied to Verma modules for restricted rational Cherednik algebras and it allows us to answer several questions posed by Gordon in some specific cases. We could determine the decomposition matrices of the Verma modules, the graded G-module structure of the simple modules, and the Calogero-Moser families of the generic restricted rational Cherednik algebra for around half of the exceptional complex reflection groups. In this way we could also confirm Martino's conjecture for several exceptional complex reflection groups.Comment: Final version to appear in LMS J. Comput. Math. 41 pages, 3 ancillary files. CHAMP is available at http://thielul.github.io/CHAMP/. All results are listed explicitly in the ancillary PDF document (currently 935 pages). Please check the website for further update

Algorithms for graded injective resolutions and local cohomology over semigroup rings

AbstractLet Q be an affine semigroup generating Zd, and fix a finitely generated Zd-graded module M over the semigroup algebra k[Q] for a field k. We provide an algorithm to compute a minimal Zd-graded injective resolution of M up to any desired cohomological degree. As an application, we derive an algorithm computing the local cohomology modules HIi(M) supported on any monomial (that is, Zd-graded) ideal I. Since these local cohomology modules are neither finitely generated nor finitely cogenerated, part of this task is defining a finite data structure to encode them

Pushforwards via scattering equations with applications to positive geometries

© 2022 The Authors. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), https://creativecommons.org/licenses/by/4.0/In this paper we explore and expand the connection between two modern descriptions of scattering amplitudes, the CHY formalism and the framework of positive geometries, facilitated by the scattering equations. For theories in the CHY family whose S-matrix is captured by some positive geometry in the kinematic space, the corresponding canonical form can be obtained as the pushforward via the scattering equations of the canonical form of a positive geometry defined in the CHY moduli space. In order to compute these canonical forms in kinematic spaces, we study the general problem of pushing forward arbitrary rational differential forms via the scattering equations. We develop three methods which achieve this without ever needing to explicitly solve any scattering equations. Our results use techniques from computational algebraic geometry, including companion matrices and the global duality of residues, and they extend the application of similar results for rational functions to rational differential forms.Peer reviewe

A Computer Algebra System for R: Macaulay2 and the m2r Package

Algebraic methods have a long history in statistics. Apart from the ubiquitous applications of linear algebra, the most visible manifestations of modern algebra in statistics are found in the young field of algebraic statistics, which brings tools from commutative algebra and algebraic geometry to bear on statistical problems. Now over two decades old, algebraic statistics has applications in a wide range of theoretical and applied statistical domains. Nevertheless, algebraic statistical methods are still not mainstream, mostly due to a lack of easy off-the-shelf implementations. In this article we debut m2r, an R package that connects R to Macaulay2 through a persistent back-end socket connection running locally or on a cloud server. Topics range from basic use of m2r to applications and design philosophy

Conormal Spaces and Whitney Stratifications

We describe a new algorithm for computing Whitney stratifications of complex projective varieties. The main ingredients are (a) an algebraic criterion, due to L\^e and Teissier, which reformulates Whitney regularity in terms of conormal spaces and maps, and (b) a new interpretation of this conormal criterion via primary decomposition, which can be practically implemented on a computer. We show that this algorithm improves upon the existing state of the art by several orders of magnitude, even for relatively small input varieties. En route, we introduce related algorithms for efficiently stratifying affine varieties, flags on a given variety, and algebraic maps.Comment: There is an error in the published version of the article (Found Comput Math, 2022) which has been fixed in this update. Section 3 is entirely new, but the downstream results Sections 4-6 remain largely the same. We have also updated the Runtimes and Complexity estimates in Section 7. The def. of the integral closure of an ideal has also been correcte