Algorithmic Elimination Methods
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Using Elimination to Describe Maxwell Curves Lucas P
Louisiana State University LSU Digital Commons LSU Master's Theses Graduate School 2006 Using elimination to describe Maxwell curves Lucas P. Beverlin Louisiana State University and Agricultural and Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_theses Part of the Applied Mathematics Commons Recommended Citation Beverlin, Lucas P., "Using elimination to describe Maxwell curves" (2006). LSU Master's Theses. 2879. https://digitalcommons.lsu.edu/gradschool_theses/2879 This Thesis is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Master's Theses by an authorized graduate school editor of LSU Digital Commons. For more information, please contact [email protected]. USING ELIMINATION TO DESCRIBE MAXWELL CURVES A Thesis Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial ful¯llment of the requirements for the degree of Master of Science in The Department of Mathematics by Lucas Paul Beverlin B.S., Rose-Hulman Institute of Technology, 2002 August 2006 Acknowledgments This dissertation would not be possible without several contributions. I would like to thank my thesis advisor Dr. James Madden for his many suggestions and his patience. I would also like to thank my other committee members Dr. Robert Perlis and Dr. Stephen Shipman for their help. I would like to thank my committee in the Experimental Statistics department for their understanding while I completed this project. And ¯nally I would like to thank my family for giving me the chance to achieve a higher education. -
Effective Noether Irreducibility Forms and Applications*
Appears in Journal of Computer and System Sciences, 50/2 pp. 274{295 (1995). Effective Noether Irreducibility Forms and Applications* Erich Kaltofen Department of Computer Science, Rensselaer Polytechnic Institute Troy, New York 12180-3590; Inter-Net: [email protected] Abstract. Using recent absolute irreducibility testing algorithms, we derive new irreducibility forms. These are integer polynomials in variables which are the generic coefficients of a multivariate polynomial of a given degree. A (multivariate) polynomial over a specific field is said to be absolutely irreducible if it is irreducible over the algebraic closure of its coefficient field. A specific polynomial of a certain degree is absolutely irreducible, if and only if all the corresponding irreducibility forms vanish when evaluated at the coefficients of the specific polynomial. Our forms have much smaller degrees and coefficients than the forms derived originally by Emmy Noether. We can also apply our estimates to derive more effective versions of irreducibility theorems by Ostrowski and Deuring, and of the Hilbert irreducibility theorem. We also give an effective estimate on the diameter of the neighborhood of an absolutely irreducible polynomial with respect to the coefficient space in which absolute irreducibility is preserved. Furthermore, we can apply the effective estimates to derive several factorization results in parallel computational complexity theory: we show how to compute arbitrary high precision approximations of the complex factors of a multivariate integral polynomial, and how to count the number of absolutely irreducible factors of a multivariate polynomial with coefficients in a rational function field, both in the complexity class . The factorization results also extend to the case where the coefficient field is a function field. -
Theory of Equations
CHAPTER I - THEORY OF EQUATIONS DEFINITION 1 A function defined by n n-1 f ( x ) = p 0 x + p 1 x + ... + p n-1 x + p n where p 0 ≠ 0, n is a positive integer or zero and p i ( i = 0,1,2...,n) are fixed complex numbers, is called a polynomial function of degree n in the indeterminate x . A complex number a is called a zero of a polynomial f ( x ) if f (a ) = 0. Theorem 1 – Fundamental Theorem of Algebra Every polynomial function of degree greater than or equal to 1 has atleast one zero. Definition 2 n n-1 Let f ( x ) = p 0 x + p 1 x + ... + p n-1 x + p n where p 0 ≠ 0, n is a positive integer . Then f ( x ) = 0 is a polynomial equation of nth degree. Definition 3 A complex number a is called a root ( solution) of a polynomial equation f (x) = 0 if f ( a) = 0. THEOREM 2 – DIVISION ALGORITHM FOR POLYNOMIAL FUNCTIONS If f(x) and g( x ) are two polynomial functions with degree of g(x) is greater than or equal to 1 , then there are unique polynomials q ( x) and r (x ) , called respectively quotient and reminder , such that f (x ) = g (x) q(x) + r (x) with the degree of r (x) less than the degree of g (x). Theorem 3- Reminder Theorem If f (x) is a polynomial , then f(a ) is the remainder when f(x) is divided by x – a . Theorem 4 Every polynomial equation of degree n ≥ 1 has exactly n roots. -
Etienne Bézout on Elimination Theory Erwan Penchevre
Etienne Bézout on elimination theory Erwan Penchevre To cite this version: Erwan Penchevre. Etienne Bézout on elimination theory. 2017. hal-01654205 HAL Id: hal-01654205 https://hal.archives-ouvertes.fr/hal-01654205 Preprint submitted on 3 Dec 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Etienne Bézout on elimination theory ∗ Erwan Penchèvre Bézout’s name is attached to his famous theorem. Bézout’s Theorem states that the degree of the eliminand of a system a n algebraic equations in n unknowns, when each of the equations is generic of its degree, is the product of the degrees of the equations. The eliminand is, in the terms of XIXth century algebra1, an equation of smallest degree resulting from the elimination of (n − 1) unknowns. Bézout demonstrates his theorem in 1779 in a treatise entitled Théorie générale des équations algébriques. In this text, he does not only demonstrate the theorem for n > 2 for generic equations, but he also builds a classification of equations that allows a better bound on the degree of the eliminand when the equations are not generic. This part of his work is difficult: it appears incomplete and has been seldom studied. -
Simple Proof of the Main Theorem of Elimination Theory in Algebraic Geometry
SIMPLE PROOF OF THE MAIN THEOREM OF ELIMINATION THEORY IN ALGEBRAIC GEOMETRY Autor(en): Cartier, P. / Tate, J. Objekttyp: Article Zeitschrift: L'Enseignement Mathématique Band (Jahr): 24 (1978) Heft 1-2: L'ENSEIGNEMENT MATHÉMATIQUE PDF erstellt am: 10.10.2021 Persistenter Link: http://doi.org/10.5169/seals-49707 Nutzungsbedingungen Die ETH-Bibliothek ist Anbieterin der digitalisierten Zeitschriften. Sie besitzt keine Urheberrechte an den Inhalten der Zeitschriften. Die Rechte liegen in der Regel bei den Herausgebern. Die auf der Plattform e-periodica veröffentlichten Dokumente stehen für nicht-kommerzielle Zwecke in Lehre und Forschung sowie für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und den korrekten Herkunftsbezeichnungen weitergegeben werden. Das Veröffentlichen von Bildern in Print- und Online-Publikationen ist nur mit vorheriger Genehmigung der Rechteinhaber erlaubt. Die systematische Speicherung von Teilen des elektronischen Angebots auf anderen Servern bedarf ebenfalls des schriftlichen Einverständnisses der Rechteinhaber. Haftungsausschluss Alle Angaben erfolgen ohne Gewähr für Vollständigkeit oder Richtigkeit. Es wird keine Haftung übernommen für Schäden durch die Verwendung von Informationen aus diesem Online-Angebot oder durch das Fehlen von Informationen. Dies gilt auch für Inhalte Dritter, die über dieses Angebot zugänglich sind. Ein Dienst der ETH-Bibliothek ETH Zürich, Rämistrasse 101, 8092 Zürich, Schweiz, www.library.ethz.ch http://www.e-periodica.ch A SIMPLE PROOF OF THE MAIN THEOREM OF ELIMINATION THEORY IN ALGEBRAIC GEOMETRY by P. Cartier and J. Tate Summary The purpose of this note is to provide a simple proof (which we believe to be new) for the weak zero theorem in the case of homogeneous polynomials. -
Triangle Identities Via Elimination Theory
Triangle Identities via Elimination Theory Dam Van Nhi and Luu Ba Thang Abstract The aim of this paper is to present some new identities via resultants and elimination theory, by presenting the sidelengths, the lengths of the altitudes and the exradii of a triangle as roots of cubic polynomials. 1 Introduction Finding identities between elements of a triangle represents a classical topic in elementary geometry. In Recent Advances in Geometric Inequalities [3], Mitrinovic et al presents the sidelengths, the lengths of the altitudes and the various radii as roots of cubic polynomials, and collects many interesting equalities and inequalities between these quantities. As far as we know, almost of them are built using only elementary algebra and geometry. In this paper, we use transformations of rational functions and resultants to build similar nice equalities and inequalities might be difficult to prove with different methods. 2 Main results To fix notations, suppose we are given a triangle ∆ABC with sidelengh a; b; c. Denote the radius of the circumcircle by R, the radius of incircle by r, the area of ∆ABC by S, the semiperimeter as p, the radii of the excircles as r1; r2; r3, and the altitudes from sides a; b and c, respectively, as ha; hb and hc. We have the results from [3, Chapter 1]. Theorem 1. [3] Using the above notations, (i) a; b; c are the roots of the cubic polynomial x3 − 2px2 + (p2 + r2 + 4Rr)x − 4Rrp: (1) (ii) ha; hb; hc are the roots of the cubic polynomial S2 + 4Rr3 + r4 2S2 2S2 x3 − x2 + x − : (2) 2Rr2 Rr R (iii) r1; r2; r3 are the roots of the cubic polynomial x3 − (4R + r)x2 + p2x − p2r: (3) Next, we present how to use transformations of rational functions to build new geometric equalities ax + b and inequalities. -
Unit 3 Cubic and Biquadratic Equations
UNIT 3 CUBIC AND BIQUADRATIC EQUATIONS Structure 3.1 Introduction Objectives 3.2 Let Us Recall Linear Equations Quadratic Equations 3.3 Cubic Equations Cardano's Solution Roots And Their Relation With Coefficients 3.4 Biquadratic Equations Ferrari's Solution Descartes' Solution Roots And Their Relation With Coeftlcients 3.5 Summary - 3.1 INTRODUCTION In this unit we will look at an aspect of algebra that has exercised the minds of several ~nntllematicians through the ages. We are talking about the solution of polynomial equations over It. The ancient Hindu, Arabic and Babylonian mathematicians had discovered methods ni solving linear and quadratic equations. The ancient Babylonians and Greeks had also discovered methods of solviilg some cubic equations. But, as we have said in Unit 2, they had llol thought of complex iiumbers. So, for them, a lot of quadratic and cubic equations had no solutioit$. hl the 16th century various Italian mathematicians were looking into the geometrical prob- lein of trisectiiig an angle by straight edge and compass. In the process they discovered a inethod for solviilg the general cubic equation. This method was divulged by Girolanlo Cardano, and hence, is named after him. This is the same Cardano who was the first to iiilroduce coinplex numbers into algebra. Cardano also publicised a method developed by his contemporary, Ferrari, for solving quartic equations. Lake, in the 17th century, the French mathematician Descartes developed another method for solving 4th degree equrtioiis. In this unit we will acquaint you with the solutions due to Cardano, Ferrari and Descartes. Rut first we will quickly cover methods for solving linear and quadratic equations. -
Elimination Theory in Differential and Difference Algebra
J Syst Sci Complex (2019) 32: 287–316 Elimination Theory in Differential and Difference Algebra∗ LI Wei · YUAN Chun-Ming Dedicated to the memories of Professor Wen-Ts¨un Wu DOI: 10.1007/s11424-019-8367-x Received: 15 October 2018 / Revised: 10 December 2018 c The Editorial Office of JSSC & Springer-Verlag GmbH Germany 2019 Abstract Elimination theory is central in differential and difference algebra. The Wu-Ritt character- istic set method, the resultant and the Chow form are three fundamental tools in the elimination theory for algebraic differential or difference equations. In this paper, the authors mainly present a survey of the existing work on the theory of characteristic set methods for differential and difference systems, the theory of differential Chow forms, and the theory of sparse differential and difference resultants. Keywords Differential Chow forms, differential resultants, sparse differential resultants, Wu-Ritt characteristic sets. 1 Introduction Algebraic differential equations and difference equations frequently appear in numerous mathematical models and are hot research topics in many different areas. Differential algebra, founded by Ritt and Kolchin, aims to study algebraic differential equations in a way similar to how polynomial equations are studied in algebraic geometry[1, 2]. Similarly, difference algebra, founded by Ritt and Cohn, mainly focus on developing an algebraic theory for algebraic differ- ence equations[3]. Elimination theory, starting from Gaussian elimination, forms a central part of both differential and difference algebra. For the elimination of unknowns in algebraic differential and difference equations, there are several fundamental approaches, for example, the Wu-Ritt characteristic set theory, the theory of Chow forms, and the theory of resultants. -
Some Lectures on Algebraic Geometry∗
Some lectures on Algebraic Geometry∗ V. Srinivas 1 Affine varieties Let k be an algebraically closed field. We define the affine n-space over k n n to be just the set k of ordered n-tuples in k, and we denote it by Ak (by 0 convention we define Ak to be a point). If x1; : : : ; xn are the n coordinate n functions on Ak , any polynomial f(x1; : : : ; xn) in x1; : : : ; xn with coefficients n in k yields a k-valued function Ak ! k, which we also denote by f. Let n A(Ak ) denote the ring of such functions; we call it the coordinate ring of n Ak . Since k is algebraically closed, and in particular is infinite, we easily see n that x1; : : : ; xn are algebraically independent over k, and A(Ak ) is hence a 0 polynomial ring over k in n variables. The coordinate ring of Ak is just k. n Let S ⊂ A(Ak ) be a collection of polynomials. We can associate to it n an algebraic set (or affine variety) in Ak defined by n V (S) := fx 2 Ak j f(x) = 0 8 f 2 Sg: This is called the variety defined by the collection S. If I =< S >:= ideal in k[x1; : : : ; xn] generated by elements of S, then we clearly have V (S) = V (I). Also, if f is a polynomial such that some power f m vanishes at a point P , then so does f itself; hence if p m I := ff 2 k[x1; : : : ; xn] j f 2 I for some m > 0g p is the radical of I, then V (I) = V ( I). -
ON the CASUS IRREDUCIBILIS of SOLVING the CUBIC EQUATION Jay Villanueva Florida Memorial University Miami, FL 33055 Jvillanu@Fmu
ON THE CASUS IRREDUCIBILIS OF SOLVING THE CUBIC EQUATION Jay Villanueva Florida Memorial University Miami, FL 33055 [email protected] I. Introduction II. Cardan’s formulas A. 1 real, 2 complex roots B. Multiple roots C. 3 real roots – the casus irreducibilis III. Examples IV. Significance V. Conclusion ******* I. Introduction We often need to solve equations as teachers and researchers in mathematics. The linear and quadratic equations are easy. There are formulas for the cubic and quartic equations, though less familiar. There are no general methods to solve the quintic and other higher order equations. When we deal with the cubic equation one surprising result is that often we have to express the roots of the equation in terms of complex numbers although the roots are real. For example, the equation – 4 = 0 has all roots real, yet when we use the formula we get . This root is really 4, for, as Bombelli noted in 1550, and , and therefore This is one example of the casus irreducibilis on solving the cubic equation with three real roots. 205 II. Cardan’s formulas The quadratic equation with real coefficients, has the solutions (1) . The discriminant < 0, two complex roots (2) ∆ real double root > 0, two real roots. The (monic) cubic equation can be reduced by the transformation to the form where (3) Using the abbreviations (4) and , we get Cardans’ formulas (1545): (5) The complete solutions of the cubic are: (6) The roots are characterized by the discriminant (7) < 0, one real, two complex roots = 0, multiple roots > 0, three real roots. -
MA3D5 Galois Theory
MA3D5 Galois theory Miles Reid Jan{Mar 2004 printed Jan 2014 Contents 1 The theory of equations 3 1.1 Primitive question . 3 1.2 Quadratic equations . 3 1.3 The remainder theorem . 4 1.4 Relation between coefficients and roots . 5 1.5 Complex roots of 1 . 7 1.6 Cubic equations . 9 1.7 Quartic equations . 10 1.8 The quintic is insoluble . 11 1.9 Prerequisites and books . 13 Exercises to Chapter 1 . 14 2 Rings and fields 18 2.1 Definitions and elementary properties . 18 2.2 Factorisation in Z ......................... 21 2.3 Factorisation in k[x] ....................... 23 2.4 Factorisation in Z[x], Eisenstein's criterion . 28 Exercises to Chapter 2 . 32 3 Basic properties of field extensions 35 3.1 Degree of extension . 35 3.2 Applications to ruler-and-compass constructions . 40 3.3 Normal extensions . 46 3.4 Application to finite fields . 51 1 3.5 Separable extensions . 53 Exercises to Chapter 3 . 56 4 Galois theory 60 4.1 Counting field homomorphisms . 60 4.2 Fixed subfields, Galois extensions . 64 4.3 The Galois correspondences and the Main Theorem . 68 4.4 Soluble groups . 73 4.5 Solving equations by radicals . 76 Exercises to Chapter 4 . 80 5 Additional material 84 5.1 Substantial examples with complicated Gal(L=k) . 84 5.2 The primitive element theorem . 84 5.3 The regular element theorem . 84 5.4 Artin{Schreier extensions . 84 5.5 Algebraic closure . 85 5.6 Transcendence degree . 85 5.7 Rings of invariants and quotients in algebraic geometry . -
DWCRTTPORS *Algebra, *Tnstr:Uction, Mathematical Concepts, *Mathematicq, Mathematics Education, *Secondary School Mathematics
DOCUMENT RESUME ED 035 974 24 SE 007 912 AUTHOR nixon, Lyle J. TTmLE An Analysis of Certain Aspects of Approximation Theory as Related to Mathematical Instruction in. Algebra. Final Report. TmsTTTUmTON Kansas state Univ., Manhattan. MOONS AGENCY Office of Education (DHEW) , Washington, D.C. Bureau of Research. W17,ATT NO RP-8-F-03n PUP DAV' Aug 6() (1PANm OFG-6-9-008030-0035(097) NOTr 99D. "UPS DRICP ;?)RS price MP-s0.50 HC-4;3.05 DWCRTTPORS *Algebra, *Tnstr:uction, Mathematical Concepts, *Mathematicq, Mathematics Education, *Secondary School mathematics AsSTRACT This study is conc9rned with certain aspects of approximation theory which can he introduced into '.he mathematics curriculum at the secondary school level. The investigation examines existing literature in mathematics which relates to this subject in an effort to determine what is available in the way of mathematical concepts pertinent to this study. As a result of the literature review the material collected has been arranged in a structured mathematical form and existing mathematical theory has been extended to make the material useful to instructional problems in high school algebra. The results of the study are found in the expository material which comprises the ma-ior portion of this report. This material contains ideas of approximation theory which relate to elementary algebra. Also, included is a collection of references for this material. The report concludes that certain techniques of approximating a root of a polynomial by finding roots of a derived polynomial can be presented in a manner which is suitable for courses in elementary algebra. (DI') EDUCATION A WELFARE U.S.