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Partial Derivatives in Arithmetic Complexity and Beyond Full Text Available At Full text available at: http://dx.doi.org/10.1561/0400000043 Partial Derivatives in Arithmetic Complexity and Beyond Full text available at: http://dx.doi.org/10.1561/0400000043 Partial Derivatives in Arithmetic Complexity and Beyond Xi Chen Columbia University USA xichen@cs.columbia.edu Neeraj Kayal Microsoft Research India neeraka@microsoft.com Avi Wigderson Institute for Advanced Study USA avi@ias.edu Boston { Delft Full text available at: http://dx.doi.org/10.1561/0400000043 Foundations and Trends R in Theoretical Computer Science Published, sold and distributed by: now Publishers Inc. PO Box 1024 Hanover, MA 02339 USA Tel. +1-781-985-4510 www.nowpublishers.com sales@nowpublishers.com Outside North America: now Publishers Inc. PO Box 179 2600 AD Delft The Netherlands Tel. +31-6-51115274 The preferred citation for this publication is X. Chen, N. Kayal and A. Wigder- son, Partial Derivatives in Arithmetic Complexity and Beyond, Foundations and Trends R in Theoretical Computer Science, vol 6, nos 1{2, pp 1{138, 2010 ISBN: 978-1-60198-480-7 c 2011 X. Chen, N. Kayal and A. Wigderson All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, mechanical, photocopying, recording or otherwise, without prior written permission of the publishers. Photocopying. In the USA: This journal is registered at the Copyright Clearance Cen- ter, Inc., 222 Rosewood Drive, Danvers, MA 01923. Authorization to photocopy items for internal or personal use, or the internal or personal use of specific clients, is granted by now Publishers Inc. for users registered with the Copyright Clearance Center (CCC). The `services' for users can be found on the internet at: www.copyright.com For those organizations that have been granted a photocopy license, a separate system of payment has been arranged. Authorization does not extend to other kinds of copy- ing, such as that for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. In the rest of the world: Permission to pho- tocopy must be obtained from the copyright owner. Please apply to now Publishers Inc., PO Box 1024, Hanover, MA 02339, USA; Tel. +1-781-871-0245; www.nowpublishers.com; sales@nowpublishers.com now Publishers Inc. has an exclusive license to publish this material worldwide. Permission to use this content must be obtained from the copyright license holder. Please apply to now Publishers, PO Box 179, 2600 AD Delft, The Netherlands, www.nowpublishers.com; e-mail: sales@nowpublishers.com Full text available at: http://dx.doi.org/10.1561/0400000043 Foundations and Trends R in Theoretical Computer Science Volume 6 Issues 1{2, 2010 Editorial Board Editor-in-Chief: Madhu Sudan Department of CS and EE MIT, Stata Center, Room G640 32 Vassar Street, Cambridge MA 02139, USA madhu@mit.edu Editors Bernard Chazelle (Princeton) Oded Goldreich (Weizmann Inst.) Shafi Goldwasser (MIT and Weizmann Inst.) Jon Kleinberg (Cornell University) L´aszl´oLov´asz(Microsoft Research) Christos Papadimitriou (UC. Berkeley) Prabhakar Raghavan (Yahoo! Research) Peter Shor (MIT) Madhu Sudan (MIT) Eva´ Tardos (Cornell University) Avi Wigderson (IAS) Full text available at: http://dx.doi.org/10.1561/0400000043 Editorial Scope Foundations and Trends R in Theoretical Computer Science will publish survey and tutorial articles in the following topics: • Algorithmic game theory • Computational Number Theory • Computational algebra • Cryptography and information • Computational aspects of security combinatorics and graph theory • Data structures • Computational aspects of • Database theory communication • Design and analysis of algorithms • Computational biology • Distributed computing • Computational complexity • Information retrieval • Computational geometry • Operations Research • Computational learning • Parallel algorithms • Computational Models and • Quantum Computation Complexity • Randomness in Computation Information for Librarians Foundations and Trends R in Theoretical Computer Science, 2010, Volume 6, 4 issues. ISSN paper version 1551-305X. ISSN online version 1551-3068. Also available as a combined paper and online subscription. Full text available at: http://dx.doi.org/10.1561/0400000043 Foundations and Trends R in Theoretical Computer Science Vol. 6, Nos. 1{2 (2010) 1{138 c 2011 X. Chen, N. Kayal and A. Wigderson DOI: 10.1561/0400000043 Partial Derivatives in Arithmetic Complexity and Beyond Xi Chen1, Neeraj Kayal2 and Avi Wigderson3 1 Columbia University, New York, NY 10027, USA, xichen@cs.columbia.edu 2 Microsoft Research, Bangalore, 560080, India, neeraka@microsoft.com 3 Institute for Advanced Study, Princeton, NJ 08540, USA, avi@ias.edu Abstract How complex is a given multivariate polynomial? The main point of this survey is that one can learn a great deal about the structure and complexity of polynomials by studying (some of) their partial deriva- tives. The bulk of the survey shows that partial derivatives provide essential ingredients in proving both upper and lower bounds for com- puting polynomials by a variety of natural arithmetic models. We will also see applications which go beyond computational complexity, where partial derivatives provide a wealth of structural information about polynomials (including their number of roots, reducibility and internal symmetries), and help us solve various number theoretic, geometric, and combinatorial problems. Full text available at: http://dx.doi.org/10.1561/0400000043 Contents 1 Introduction 1 1.1 Motivation 1 1.2 Arithmetic Circuits 4 1.3 Formal Derivatives and Their Properties 9 Part I: Structure 13 2 Symmetries of a Polynomial 17 2.1 Proof of Lemma 2.1 21 2.2 Proof of Lemma 2.4 25 3 Algebraic Independence 27 4 Polynomials with High Arithmetic Complexity 33 5 Bezout's Theorem 37 5.1 A Property of Isolated Roots 38 5.2 Proof of Theorem 5.2 39 5.3 Proof of Lemma 5.3 42 5.4 Bezout's Theorem over Finite Fields and Rings 43 ix Full text available at: http://dx.doi.org/10.1561/0400000043 5.5 A Brief Introduction to the p-adic Numbers, Norm, and Metric 44 5.6 Sketch of Proof of Theorem 5.6 46 6 Algebraic Extractors and the Jacobian Conjecture 49 6.1 The Jacobian Conjecture 50 7 The \Joints Conjecture" Resolved 53 8 The Stepanov Method 55 8.1 Weil's Theorem on Rational Points on Curves 56 8.2 A High-level Description of the Stepanov Method 62 8.3 Formal Proof of Weil's Theorem 66 8.4 The Heath-Brown and Mit'kin Estimates 70 Part II: Lower Bounds 73 9 General Arithmetic Circuits 75 10 Sums of Powers of Linear Forms 85 10.1 Arithmetic Circuits with Addition and Powering Gates 88 10.2 Depth-2 Symmetric Arithmetic Circuits 88 11 Depth-3 Arithmetic Circuits 91 12 Arithmetic Formulae 95 12.1 Cover Sets and Measure Functions 97 12.2 A Constant-depth Lower Bound 98 Full text available at: http://dx.doi.org/10.1561/0400000043 13 Projections of Determinant to Permanent 103 Part III: Algorithms 109 14 Identity Testing 111 14.1 POLYDEP and its Connection to Identity Testing 112 14.2 Basic Properties of POLYDEP 113 14.3 The Algorithm 114 15 Absolute Irreducibility Testing 119 15.1 Notation 120 15.2 Consequences 122 16 Polynomial Equivalence Testing 125 16.1 Algorithm for Minimizing the Number of Variables 128 16.2 Equivalence to a Sum of Powers 130 16.3 Equivalence to an Elementary Symmetric Polynomial 132 Acknowledgments 139 References 141 Full text available at: http://dx.doi.org/10.1561/0400000043 1 Introduction 1.1 Motivation Polynomials are perhaps the most important family of functions in mathematics. They feature in celebrated results from both antiquity and modern times, like the unsolvability by radicals of polynomials of degree ≥5 of Abel and Galois, and Wiles' proof of Fermat's \last theorem." In computer science they feature in, for example, error- correcting codes and probabilistic proofs, among many applications. The manipulation of polynomials is essential in numerous applications of linear algebra and symbolic computation. This survey is devoted mainly to the study of polynomials from a computational perspective. The books [9, 10, 86] and the recent survey [74] provide wide coverage of the area. Given a polynomial over a field, a natural question to ask is how complex it is? A natural way to compute polynomials is via a sequence of arithmetic operations, for example, by an arithmetic circuit, as shown in Figure 1.1 (formal definitions will be given in Section 1.2). One definition of how complex a polynomial is can be the size of the smallest arithmetic circuit computing it. A weaker model, often employed by mathematicians, is that of a formula (in which the underlying circuit 1 Full text available at: http://dx.doi.org/10.1561/0400000043 2 Introduction structure must be a tree), and another definition of complexity may be the formula size. There are many ways to compute a given polynomial. For example, f(x1;x2) = x1 × (x1 + x2) + x2 × (x1 + x2) = (x1 + x2) × (x1 + x2) are two formulae for the same polynomial f, the first requiring 5 oper- ations and the second only 3 operations. Finding the optimal circuit or formula computing a given polynomial is a challenging task, and even estimating that minimum size by giving upper and lower bounds is very difficult. Of course, the same is also true for the study of Boolean functions and their complexity (with respect to Boolean circuits and formulae, or Turing machines), but in the Boolean case we have a bet- ter understanding of that difficulty (via results on relativization by Baker et al. [5], natural proofs due to Razborov and Rudich [64], and algebrization due to Aaronson and Wigderson [1]). For the arithmetic setting, which is anyway more structured, there seem to be more hope for progress. Proving lower bounds for the complexity of polynomials has been one of the most challenging problems in theoretical computer science.
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