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Poisson traces, D-modules, and symplectic resolutions

LETTERS IN MATHEMATICAL PHYSICS 2018年第3期108卷 633-678页

作者： Etingof, Pavel Schedler, Travis MIT 77 Massachusetts Ave Cambridge MA 02139 USA Imperial Coll London London England

We survey the theory of Poisson traces (or zeroth Poisson homology) developed by the authors in a series of recent papers. The goal is to understand this subtle invariant of (singular) Poisson varieties, conditions for it to be finite-dimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations. The main technique is the study of a canonical D-module on the variety. In the case the variety has finitely many symplectic leaves (such as for symplectic singularities and Hamiltonian reductions of symplectic vector spaces by reductive groups), the D-module is holonomic, and hence, the space of Poisson traces is finite-dimensional. As an application, there are finitely many irreducible finite-dimensional representations of every quantization of the variety. Conjecturally, the D-module is the pushforward of the canonical D-module under every symplectic resolution of singularities, which implies that the space of Poisson traces is dual to the top cohomology of the resolution. We explain many examples where the conjecture is proved, such as symmetric powers of du Val singularities and symplectic surfaces and Slodowy slices in the nilpotent cone of a semisimple Lie algebra. We compute the D-module in the case of surfaces with isolated singularities and show it is not always semisimple. We also explain generalizations to arbitrary Lie algebras of vector fields, connections to the Bernstein-Sato polynomial, relations to two-variable special polynomials such as Kostka polynomials and Tutte polynomials, and a conjectural relationship with deformations of symplectic resolutions. In the appendix we give a brief recollection of the theory of D-modules on singular varieties that we require.

关键词： Hamiltonian flow Complete intersections Milnor number D-modules Poisson homology Poisson varieties Poisson homology Poisson traces Milnor fibration Calabi-Yau varieties Deformation quantization Kostka polynomials Symplectic resolutions Twistor deformations

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The theory of besov functional calculus: Developments and applications to semigroups

arXiv

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arXiv 2019年

作者： Batty, Charles Gomilko, Alexander Tomilov, Yuri St. John's College University of Oxford OxfordOX1 3JP United Kingdom Faculty of Mathematics and Computer Science Nicolas Copernicus University Chopin Street 12/18 Torun87-100 Poland Institute of Mathematics Polish Academy of Sciences Sniadeckich ´ 8 Warsaw00-956 Poland

We extend and deepen the theory of functional calculus for semigroup generators, based on the algebra B of analytic Besov functions, which we initiated in a previous paper. In particular, we show that our construction of the calculus is optimal in several natural senses. Moreover, we clarify the structure of B and identify several important subspaces in practical terms. This leads to new spectral mapping theorems for operator semigroups and to wide generalisations of a number of basic results from semigroup theory. Copyright © 2019, The Authors. All rights reserved.

关键词： algebra

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The Fundamental Theorem of algebra in ACL2 15

The Fundamental Theorem of Algebra in ACL2

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15th International Workshop on the ACL2 Theorem Prover and Its applications

作者： Gamboa, Ruben Cowles, John Univ Wyoming Laramie WY 82071 USA

We report on a verification of the Fundamental Theorem of algebra in ACL2(r). The proof consists of four parts. First, continuity for both complex-valued and real-valued functions of complex numbers is defined, and it is shown that continuous functions from the complex to the real numbers achieve a minimum value over a closed square region. An important case of continuous real-valued, complex functions results from taking the traditional complex norm of a continuous complex function. We think of these continuous functions as having only one (complex) argument, but in ACL2(r) they appear as functions of two arguments. The extra argument is a "context", which is uninterpreted. For example, it could be other arguments that are held fixed, as in an exponential function which has a base and an exponent, either of which could be held fixed. Second, it is shown that complex polynomials are continuous, so the norm of a complex polynomial is a continuous real-valued function and it achieves its minimum over an arbitrary square region centered at the origin. This part of the proof benefits from the introduction of the "context" argument, and it illustrates an innovation that simplifies the proofs of classical properties with unbound parameters. Third, we derive lower and upper bounds on the norm of non-constant polynomials for inputs that are sufficiently far away from the origin. This means that a sufficiently large square can be found to guarantee that it contains the global minimum of the norm of the polynomial. Fourth, it is shown that if a given number is not a root of a non-constant polynomial, then it cannot be the global minimum. Finally, these results are combined to show that the global minimum must be a root of the polynomial. This result is part of a larger effort in the formalization of complex polynomials in ACL2(r).

关键词： polynomials

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Standard Lattices of Compatibly Embedded Finite Fields

arXiv

引用

arXiv 2019年

作者： de Feo, Luca Randriam, Hugues Rousseau, Édouard Université Paris Saclay UVSQ LMV LTCI Télécom ParisTech LTCI Télécom ParisTech Université Paris Saclay UVSQ LMV

Lattices of compatibly embedded finite fields are useful in computer algebra systems for managing many extensions of a finite field Fp at once. They can also be used to represent the algebraic closure F¯p , and to represent all finite fields in a standard manner. The most well known constructions are Conway polynomials, and the Bosma–Cannon–Steel framework used in Magma. In this work, leveraging the theory of the Lenstra-Allombert isomorphism algorithm, we generalize both at the same time. Compared to Conway polynomials, our construction defines a much larger set of field extensions from a small pre-computed table;however it is provably as inefficient as Conway polynomials if one wants to represent all field extensions, and thus yields no asymptotic improvement for representing F¯p . Compared to Bosma–Cannon–Steel lattices, it is considerably more efficient both in computation time and storage: all algorithms have at worst quadratic complexity, and storage is linear in the number of represented field extensions and their degrees. Our implementation written in C/Flint/Julia/Nemo shows that our construction in indeed practical. Copyright © 2019, The Authors. All rights reserved.

关键词： polynomials

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PolyCleaner: Clean your polynomials before Backward Rewriting to Verify Million-gate Multipliers 18

PolyCleaner: Clean your Polynomials before Backward Rewritin...

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37th IEEE/ACM International Conference on computer-Aided Design (ICCAD)

作者： Mahzoon, Alireza Grosse, Daniel Drechsler, Rolf Univ Bremen Inst Comp Sci Bremen Germany DFKI GmbH Cyber Phys Syst Bremen Germany

ISBN: (纸本)9781450359504

Nowadays, a variety of multipliers are used in different computationally intensive industrial applications. Most of these multipliers are highly parallelized and structurally complex. Therefore, the existing formal verification techniques fail to verify them. In recent years, formal multiplier verification based on Symbolic computer algebra (SCA) has shown superior results in comparison to all other existing proof techniques. However, for non-trivial architectures still a monomial explosion can be observed. A common understanding is that this is caused by redundant monomials also known as vanishing monomials. While several approaches have been proposed to overcome the explosion, the problem itself is still not fully understood. In this paper we present a new theory for the origin of vanishing monomials and how they can be handled to prevent the explosion during backward rewriting. We implement our new approach as the SCA-verifier PolyCleaner. The experimental results show the efficiency of our proposed

关键词： Logic gates Explosions Adders algebra computer architecture Databases computer science

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A theory and an Algorithm forÂ Computing Sparse Multivariate Polynomial Remainder Sequence 20th

A Theory and an Algorithm forÂ Computing Sparse Multivaria...

引用

20th International Workshop on computer algebra in Scientific Computing, CASC 2018

作者： Sasaki, Tateaki University of Tsukuba Tsukuba-shiIbaraki305-8571 Japan

ISBN: (纸本)9783319996387

This paper presents an algorithm for computing the polynomial remainder sequence (PRS) and corresponding cofactor sequences of sparse multivariate polynomials over a number field. Most conventional algorithms for computing PRSs are based on the pseudo remainder (Prem), and the celebrated subresultant theory for the PRS has been constructed on the Prem. The Prem is uneconomical for computing PRSs of sparse polynomials. Hence, in this paper, the concept of sparse pseudo remainder (spsPrem) is defined. No subresultant-like theory has been developed so far for the PRS based on spsPrem. Therefore, we develop a matrix theory for spsPrem-based PRSs. The computational formula for PRS, regardless of whether it is based on Prem or spsPrem, causes a considerable intermediate expression growth. Hence, we next propose a technique to suppress the expression growth largely. The technique utilizes the power-series arithmetic but no Hensel lifting. Simple experiments show that our technique suppresses the intermediate expression growth fairly well, if the sub-variable ordering is set suitably. © 2018, Springer Nature Switzerland AG.

关键词： polynomials

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Self-dual skew cyclic codes over Fq + uFq

arXiv

引用

arXiv 2019年

作者： Hebbache, Zineb Guenda, Kenza Tugba Özzaim, N. Özen, Mehmet Aaron Gulliver, T. Faculty of Mathematics USTHB Laboratory of Algebra and Number Theory BP 32 El Alia Bab Ezzouar Algeria Department of Mathematics Sakarya University Sakarya Turkey Department of Electrical and Computer Engineering University of Victoria PO Box 1700 STN CSC VictoriaBCV8W 2Y2 Canada

In this paper, we give conditions for the existence of Hermitian self-dual Θ−cyclic and Θ−negacyclic codes over the finite chain ring Fq + uFq. By defining a Gray map from R = Fq + uFq to F2q, we prove that the Gray images of skew cyclic codes of odd length n over R with even characteristic are equivalent to skew quasi-twisted codes of length 2n over Fq of index 2. We also extend an algorithm of Boucher and Ulmer [9] to construct self-dual skew cyclic codes based on the least common left multiples of non-commutative polynomials over Fq + uFq. Copyright © 2019, The Authors. All rights reserved.

关键词： polynomials

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The SAT+CAS Method for Combinatorial Search with applications to Best Matrices

arXiv

引用

arXiv 2019年

作者： Bright, Curtis Ðoković, Dragomir Ž. Kotsireas, Ilias Ganesh, Vijay University of Waterloo Wilfrid Laurier University

In this paper, we provide an overview of the SAT+CAS method that combines satisfiability checkers (SAT solvers) and computer algebra systems (CAS) to resolve combinatorial conjectures, and present new results vis-à-vis best matrices. The SAT+CAS method is a variant of the Davis–Putnam–Logemann–Loveland DPLL(T) architecture, where the T solver is replaced by a CAS. We describe how the SAT+CAS method has been previously used to resolve many open problems from graph theory, combinatorial design theory, and number theory, showing that the method has broad applications across a variety of fields. Additionally, we apply the method to construct the largest best matrices yet known and present new skew Hadamard matrices constructed from best matrices. We show the best matrix conjecture (that best matrices exist in all orders of the form r2 + r + 1) which was previously known to hold for r ≤ 6 also holds for r = 7. We also confirmed the results of the exhaustive searches that have been previously completed for r ≤ 6. Copyright © 2019, The Authors. All rights reserved.

关键词： number theory

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Convex Relaxations for Consensus and Non-Minimal Problems in 3D Vision

Convex Relaxations for Consensus and Non-Minimal Problems in...

引用

International Conference on computer Vision (ICCV)

作者： Thomas Probst Danda Pani Paudel Ajad Chhatkuli Luc Van Gool Computer Vision Laboratory ETH Zurich Switzerland ETH Zurich Zurich Switzerland

ISBN: (数字)9781728148038

ISBN: (纸本)9781728148045

In this paper, we formulate a generic non-minimal solver using the existing tools of polynomials Optimization Problems (POP) from computational algebraic geometry. The proposed method exploits the well known Shor's or Lasserre's relaxations, whose theoretical aspects are also discussed. Notably, we further exploit the POP formulation of non-minimal solver also for the generic consensus maximization problems in 3D vision. Our framework is simple and straightforward to implement, which is also supported by three diverse applications in 3D vision, namely rigid body transformation estimation, Non-Rigid Structure-from-Motion (NRSfM), and camera autocalibration. In all three cases, both non-minimal and consensus maximization are tested, which are also compared against the state-of-the-art methods. Our results are competitive to the compared methods, and are also coherent with our theoretical analysis. The main contribution of this paper is the claim that a good approximate solution for many polynomial problems involved in 3D vision can be obtained using the existing theory of numerical computational algebra. This claim leads us to reason about why many relaxed methods in 3D vision behave so well? And also allows us to offer a generic relaxed solver in a rather straightforward way. We further show that the convex relaxation of these polynomials can easily be used for maximizing consensus in a deterministic manner. We support our claim using several experiments for aforementioned three diverse problems in 3D vision.

关键词： Three-dimensional displays Symmetric matrices Optimization Cameras Minimization computer vision Estimation

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Spectral analysis of matrix scaling and operator scaling

arXiv

引用

arXiv 2019年

作者： Kwok, Tsz Chiu Lau, Lap Chi Ramachandran, Akshay Institute for Theoretical Computer Science Shanghai University of Finance and Economics School of Computer Science University of Waterloo

We present a spectral analysis for matrix scaling and operator scaling. We prove that if the input matrix or operator has a spectral gap, then a natural gradient flow has linear convergence. This implies that a simple gradient descent algorithm also has linear convergence under the same assumption. The spectral gap condition for operator scaling is closely related to the notion of quantum expander studied in quantum information theory. The spectral analysis also provides bounds on some important quantities of the scaling problems, such as the condition number of the scaling solution and the capacity of the matrix and operator. These bounds can be used in various applications of scaling problems, including matrix scaling on expander graphs, permanent lower bounds on random matrices, the Paulsen problem on random frames, and Brascamp-Lieb constants on random operators. In some applications, the inputs of interest satisfy the spectral condition and we prove significantly stronger bounds than the worst case bounds. Copyright © 2019, The Authors. All rights reserved.

关键词： Matrix algebra

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