# Mathematics for computer algebra - Mignotte M.

### Mathematics for computer algebra

Author: Mignotte M.Publishers: New York

Year of publication: 1992

Number of pages: 92

ISBN 0-387-97675-2

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Maurice Mignotte Universit6 Louis Pasteur Department de Math6matique 67084 Strasbourg France

Mathematics Subject Classification: 11Y05,11Y11,12D50,12Y05,13P0S, 68Q40

This book was originally published in French by the Presses Universitaires de France, 1989. The French edition is entitled Mathematigues pour Ie calculformel.

Library of Congress Cataloging-in-Publication Data Mignotte, Maurice.

[Math^matiques pour Ie calcul formel. English]

Mathematics for computer algebra / Maurice Mignotte : translated by Catherine Mignotte. p. cm.

Translation of: Mathfrnatiques pour Ie calcul formel.

Includes bibliographical references and index.

ISBN 0-387-97675-2

I. Algebra—Data processing. I. Title.

QA155.7.E4M5213 1991

512 —dc20 91-33024

Printed on acid-free paper.

© 1992 Springer-Verlag New York, Inc.

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.

The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as Understood by the Tirade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Production managed by Henry Krell; manufacturing supervised by Robert Paella.

Camera ready copy provided by the author.

Printed and bound by R.R. Donnelley and Sons, Harrisonburg, VA.

Printed in the United States of America.

987654321

ISBN 0-387-97675-2 Springer-Verlag New York Berlin Heidelberg ynnkT -* åëï þñïñ ë ñ^ã³ïëëã.ë/àã³ÿî- R/»rlin Hpidelbere New York

PREFACE

This book corresponds to a mathematical course given in 1986/87 University Louis Pasteur, Strasbourg.

This work is primarily intended for graduate students. The foil are necessary prerequisites : a few standard definitions in set theo* definition of rational integers, some elementary facts in Combine (maybe only Newton’s binomial formula), some theorems of AnaJj the level of high schools, and some elementary Algebra (basic results groups, rings, fields and linear algebra).

An important place is given to exercises. These exercises are only direct applications of the course. More often, they constitute comple to the text. Mostly, hints or references are given so that the reader ³ be able to find solutions.

Chapters one and two deal with elementary results of Number T for example : the euclidean algorithm, the Chinese remainder theore Fermat’s little theorem. These results are useful by themselves, bu also constitute a concrete introduction to some notions in abstract a (for example, euclidean rings, principal rings ...). Algorithms are gr arithmetical operations with long integers.

The rest of the book, chapters 3 through 7, deals with polynomia give general results on polynomials over arbitrary rings. Then polyn with complex coefficients are studied in chapter 4, including many est on the complex roots of polynomials. Some of these estimates are very in the subsequent chapters.

Chapter 5 introduces polynomials with real coefficients. The main of this chapter is the separation of the real roots of real polynomiE recall many results of the last century, which generally do not apj modern textbooks. Among them are Sturm’s method, the rules of De

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1°) Suppose that C = (cij)i<i,j<n is a matrix with elements in some commutative ring A such that there exists an integer m > n and elements a.ia, bj s in A, I < i, j < n and I < s < m satisfying the relations

m

Ci j ^ 8 bj S ) I ^ j — lb •

a—I

If I = {ii,..., in} is any subset of cardinality n of the interval {1,2,... ,m} we put

À-i ^ ] iai tj ¦ • • à» *„ and Bi = ^ ] ibi I1 ¦ ¦ ,

where the signs are given by the signatures of the permutations (as for determinants). Then, prove the formula

det (C) = ]T Ai B1,

I

where the Is run over the subsets of cardinality n of the interval {1,2 ,...,m}.

2°) Let / be monic univariate polynomial of positive degree n, with coefficients in A. For any nonnegative integer i, let Si be the sum of the ith powers of the roots of /. For any integer v, I < v < n, set

Dv = det (cfij), where CTjj = Si+j_2, I < i, j < n.

If Zi, ..., Zn are n indeterminates, put also

V(zi,..., Zn) = JJ (zi — Zj).

I <i<j<n

Using the previous question, demonstrate the formula Dv = Y V2 (Oil,..., Oii,),

where the sum is extended to all subsets {ij,..., iv} of cardinality v of the set {1,2,..., n}. Deduce the fact that / has exactly r distinct roots if, and only if Dn = ?>n_i = • • • = Dr+i = 0 but Dr ô 0.

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Chapter 4

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