SIGSAM Bull.最新文献

{"title":"The package CRACK for solving large overdetermined systems","authors":"Thomas Wolf","doi":"10.1145/1113439.1113455","DOIUrl":"https://doi.org/10.1145/1113439.1113455","url":null,"abstract":"The program CRACK is a computer algebra package written inREDUCE for the solution of over-determined systems of algebraic,ordinary or partial differential equations with at most polynomialnon-linearity. It is available as part of version 3.8 of the REDUCEsystem (dated April 2004) and the latest development version can bedownloaded from http://lie.math.brocku.ca/crack/.\u0000The purpose of this poster is to accompany a softwaredemonstration of CRACK at ISSAC 2005. The poster is supposed togive a graphical overview of CRACK, emphasizing features which arespecial to the package. Those are\u0000• a rich interface, with visualization aids forinspecting large systems, including\u0000- for each equation its properties, history and investigationsthat have already been done with the equation,\u0000- the occurrence of all derivatives of selected functions in anyequation,\u0000- a statistical overview of the system (number of equations andfunctions in dependence of number of independent variables),\u0000- a matrix display of occurrences of unknown constants/functionsin all equations,\u0000- a count of the total number of appearances of eachunknown,\u0000- the determination of not under-determined subsystems,\u0000- a listing of all sub- and sub-sub-.. cases investigated sofar, with their assumptions, number of steps and number ofsolutions,\u0000- graphical displays of size related measures of the computationdone so far;\u0000• a discussion of possibilities to trade interactivelyor automatically the speed of the solution process versus safety(avoidance of expression swell):\u0000- only length-reducing versus complete Gröbner basiscomputation steps.\u0000- substitutions in shorter equations only, i.e. only in asub-system versus substitutions in the complete system,\u0000- growth bounded substitutions versus general substitutions,\u0000- case splittings (induced by factorizations, substitutions withpotentially vanishing coefficients or adhoc case distinctions)versus Gröbner basis steps,\u0000- an investment in the length reduction of equations to reachsparse systems with multiple benefits;\u0000• algorithmic extensions which include\u0000- the ability to collect and apply syzygies which result as aby-product in the process of computing a differentialGröbner basis to integrate linear PDEs.\u0000- the treatment of inequalities: their usage, active collectionand derivation, and their constant update in an ongoing reductionprocess based on newly derived equations.\u0000- the capability added by Winfried Neun (ZIB Berlin) to run in atruly parallel mode on a beowulf cluster, recently also ported to64bit AMD processors.\u0000- post-computation procedures, especially the possibility tomerge solutions of parametric linear algebraic systems and toautomate the production of web-pages for solutions that arefound.\u0000- the ability to separate expressions with respect toindependent variables which occur polynomially but with variableexponents, leading to automatically investigated case distinctionsof exponents being equal or not;\u0000• safety enhanc","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115777910","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An approach to singularity from the extended Hensel construction","authors":"Tateaki Sasaki, D. Inaba, K. Katamachi","doi":"10.1145/1113439.1113454","DOIUrl":"https://doi.org/10.1145/1113439.1113454","url":null,"abstract":"Let <i>F</i>(<i>x,u</i>), with (<i>u</i>) = (<i>u</i><inf>1</inf>,...,<i>u</i><inf>ℓ</inf>), be an irreducible multivariate polynomial over C, having singularity at the origin, and let <i>F</i><inf>New</inf>(<i>x,u</i>) be the so-called Newton polynomial for <i>F</i>(<i>x,u</i>). The extended Hensel construction (EHC in short) of <i>F</i>(<i>x,u</i>) allows us to compute the Puiseux-series roots if ℓ = 1 and, for ℓ ≥ 2, the roots which are fractional-power series w.r.t. the (weighted) total-degree of <i>u</i><inf>1</inf>,...,<i>u</i><inf>ℓ</inf>. This paper investigates the behavior of algebraic function χ(<i>u</i>) around the origin, where χ(<i>u</i>) is a root of <i>F</i>(<i>x,u</i>), w.r.t. the variable <i>x</i>, and clarifies a close relationship between the singularity of χ(<i>u</i>) and the corresponding Hensel factor. Let the irreducible factorization of <i>F</i><inf>New</inf> in C[<i>x,u</i>] be <i>F</i><inf>New</inf>(<i>x,u</i>)^<i>m</i><inf>1</inf>...<i>H</i><inf><i>r</i></inf>(<i>x,u</i>)^<i>m</i>. It is shown that: 1) the procedure of EHC distributes the factors of the leading coefficient of <i>F</i>(<i>x,u</i>) to mutually prime factors of <i>F</i><inf>New</inf> in a unique way, and the \"scaled-root\" <i>x</i>(<i>u</i>) becomes infinity only at the zero-points of the leading coefficient of the corresponding factor of <i>F</i><inf>New</inf>(<i>x,u</i>); 2) the behavior of χ(<i>u</i>) around the origin changes singularly at zero-points of resultant(<i>H</i><inf><i>i</i></inf>, <i>H</i><inf><i>j</i></inf>) (∀<i>i</i> ≠ <i>j</i>), resultant(<i>H</i><inf>i</inf>,∂<i>H<inf>i</inf></i>/∂<i>x</i>) (<i>i</i> = 1,...,<i>r</i>), and <i>fcirc;</i><inf>ℓ</inf> (ℓ = 1,...,ρ), where <i>f</i><inf>ℓ</inf> is the sum of terms plotted at the ℓ-th vertex of bottom sides of the \"Newton polygon\". We explain these points not only theoretically but also by examples.","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121731875","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Computation with hyperexponential functions","authors":"Ziming Li, Dabin Zheng","doi":"10.1145/1113439.1113446","DOIUrl":"https://doi.org/10.1145/1113439.1113446","url":null,"abstract":"A multivariate hyperexponential function is a function whose\"logarithmic derivatives\" are rational. Examples ofhyperexponential functions include rational functions, exponentialfunctions, and hypergeometric terms. Hyperexponential functionsplay an important role in the handling of analytic andcombinatorial objects. We present a few algorithms applicable tothe manipulation of hyperexponential functions in an uniformway.\u0000Let <i>F</i> be a field of characteristic zero, onwhich derivation operators&delta;<inf>1</inf>,...,&delta;<inf>&ell;</inf>and difference operators (automorphisms)&sigma;<inf>&ell;+1</inf>,...,&sigma;<inf>m</inf> act. Let <i>E</i>be an <i>F</i>-algebra. Assume that the&delta;<inf><i>i</i></inf> for 1&le; <i>i</i> &le; &ell; and&sigma;<inf><i>j</i></inf> for&ell; + 1 &le; <i>m</i> can be extended to<i>E</i> as derivation and difference operators.Moreover, these operators commute with each other on<i>E.</i> A hyperexponential element of<i>E</i> over <i>F</i> is defined to be anonzero element <i>h</i> &isin;<i>E</i> such that\u0000&delta;<inf>1</inf>(<i>h</i>) =<i>r</i><inf>1</inf><i>h</i>,...,&delta;<inf>&ell;</inf>(<i>h</i>)=<i>r</i><inf>&ell;</inf><i>h</i>,&sigma;<inf>&ell;+1</inf>(<i>h</i>)=<i>r</i><inf>&ell;+1</inf><i>h</i>,...,&sigma;<inf><i>m</i></inf>(<i>h</i>)=<i>r</i><inf><i>m</i></inf><i>h</i>\u0000for some <i>r</i><inf>1</inf>,...,<i>r<inf>m</inf></i> &isin;<i>F</i>. These rational functions are called(rational) certificates for <i>h.</i>","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129843633","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the complexity of the D5 principle","authors":"X. Dahan, É. Schost, M. M. Maza, Wenyuan Wu, Yuzhen Xie","doi":"10.1145/1113439.1113457","DOIUrl":"https://doi.org/10.1145/1113439.1113457","url":null,"abstract":"The standard approach for computing with an algebraic number isthrough the data of its irreducible minimal polynomial over somebase field k. However, in typical tasks such as polynomialsystem solving, involving many algebraic numbers of high degree,following this approach will require using probably costlyfactorization algorithms. Della Dora, Dicrescenzo and Duvalintroduced \"dynamic evaluation\" techniques (also termed \"D5principle\") [3] as a means to compute with algebraic numbers, whileavoiding factorization. Roughly speaking, this approach leads oneto compute over direct products of field extensions of k,instead of only field extensions.\u0000In this work, we address complexity issues for basic operationsin such structures. Precisely, let\u0000[EQUATION]\u0000be a family of polynomials, called a <i>triangularset</i>, such that <i>k</i> &larr;<i>K</i> =<i>k</i>[<i>X</i><inf>1</inf>,...,<i>X<inf>n</inf></i>]/<b>T</b>is a direct product of field extensions. We write &delta; forthe dimension of <i>K</i> over <i>k</i>,which we call the <i>degree</i> of<b>T.</b> Using fast polynomial multiplication andNewton iteration for power series inverse, it is a folklore resultthat for any &epsilon; &gt; 0, the operations (+, X) in<i>K</i> can be performed in<i>c</i>^<i>n</i><inf>&epsilon;</inf>&delta;^1+&epsilon;operations in <i>k</i>, for some constant<i>c</i><inf>&epsilon;</inf>. Using afast Euclidean algorithm, a similar result easily carries over toinversion, <i>in the special case when K is afield.</i>\u0000Our main results are similar estimates for the general case,where <i>K</i> is merely a product of fields. Followingthe D5 philosophy, meeting zero-divisors in the computation willlead to <i>splitting</i> the triangular set<b>T</b> into a family thereof, defining the sameextension. Inversion is then replaced by<i>quasi-inversion:</i> a quasi-inverse [6] of&alpha; &isin; <i>K</i> is a splitting of<b>T</b>, such that &alpha; is either zero orinvertible in each component, together with the data of thecorresponding inverses.","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126265373","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A note on a nearest polynomial with a given root","authors":"S. Graillat","doi":"10.1145/1101884.1101887","DOIUrl":"https://doi.org/10.1145/1101884.1101887","url":null,"abstract":"In this paper, we consider the problem of a nearest polynomial with a given root in the complex field (the coefficients of the polynomial and the root are complex numbers). We are interested in the existence and the uniqueness of such polynomials. Then we study the problem in the real case (the coefficients of the polynomial and the root are real numbers), and in the real-complex case (the coefficients of the polynomial are real numbers and the root is a complex number). We derive new formulas for computing such polynomials.","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122792593","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"SAGE: system for algebra and geometry experimentation","authors":"W. Stein, D. Joyner","doi":"10.1145/1101884.1101889","DOIUrl":"https://doi.org/10.1145/1101884.1101889","url":null,"abstract":"SAGE is a framework for number theory, algebra, and geometry computation that is initially being designed for computing with elliptic curves and modular forms. The current implementation is primarily due to William Stein. It is open source and freely available under the terms of the GNU General Public License (GPL).","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132787421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"GUAVA: an error-correcting codes package","authors":"D. Joyner","doi":"10.1145/1101884.1101890","DOIUrl":"https://doi.org/10.1145/1101884.1101890","url":null,"abstract":"GUAVA is a GAP 4 package primarily designed for the construction and analysis of error-correcting codes authored by Jasper Cramwinckel, Erik Roijackers, Reinald Baart, Eric Minkes, Lea Ruscio, and David Joyner. Though GUAVA can call some special functions written in C, most of GUAVA is written in GAP's own interpretive language. Several algorithms that need the speed were integrated into the GAP kernel, also written in C, by Steve Linton.","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"79 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115774890","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Conjectural permutation decoding of some AG codes","authors":"D. Joyner","doi":"10.1145/1080368.1080374","DOIUrl":"https://doi.org/10.1145/1080368.1080374","url":null,"abstract":"We study the action of a finite group on the Riemann-Roch space of certain divisors on a specific hyperelliptic curve <i>X</i> defined over a finite field with \"large\" automorphism group <i>G</i>. If <i>D</i> and <i>E</i> = <i>P</i><inf>l</inf> + ... + <i>P<inf>n</inf></i> are <i>G</i>-equivariant divisors on <i>X</i> (<i>P<inf>i</inf></i> ∈ <i>X</i>(<i>F</i>)) then <i>G</i> acts on associated AG code <i>C</i> = <i>C</i>(<i>D,E</i>) by permuting coordinates. This note discusses the permutation decoding of these AG codes. The main \"results\" are conjectures regarding the complexity of the permutation decoding of these hyperelliptic codes. The open source GAP error-correcting codes package GUAVA is used to compute examples.","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"98 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116397620","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"East coast computer algebra day 2005: poster abstracts","authors":"I. Ajwa","doi":"10.1145/1080368.1080372","DOIUrl":"https://doi.org/10.1145/1080368.1080372","url":null,"abstract":"We present a system that allows wireless smartphones to be used for mathematics communication, that is, for the creation and exchange of mathematical formulas, diagrams, and text between two or more participants. The system enables two or more persons with smartphones or traditional computers to participate in a session. Each of the participants may convey textual, graphical and mathematical information to the other participants. Users can draw, edit, and label geometric shapes, send chat messages, and compose formulas. A turn taking mechanism moderates the communication. The system also supports the addition of services that can be used to provide individual users with additional functionality. Currently, a L ATEX rendering service is available to allow users to create and share mathematical formulas in typeset quality. As the use of smartphones becomes more widespread we expect that our system will be used by online communities such as the Math Forum at Drexel University. Students and teachers of all levels use the Math Forum in order to ask questions, solve problems, and pursue interests in mathematics. The service Ask Dr. Mathreceives as many as 9,000 questions a month and over 200 trained volunteers—teachers, professionals in the workplace, college students, and hobbyists—answer close to 4,000 of these directly within a few hours of their submission. Over 7,800 questions and answers have been published in public archives, with another 400,000 questions and 120,000 answers stored for research and future use. Currently the Math Forum receives about 1.7 million visits a month, and approximately 450,000 of those include the Dr. Math area. Some of the challenges of developing a system for mobile communication of high school mathematics are posed by the input and output limitations of the devices and the enormous heterogeneity of the available hardware platforms. We are using the Treo 600 as our hardware platform. Our software is written in pure Java and can potentially be run on any platform that supports the Java Virtual Machine. Our software architecture makes it easy to add new local or remote services to enhance the communication. Using Riemannian SVD for Problems in Approximate Algebra Brad Botting, Mark Giesbrecht, John May University of Waterloo Abstract. Many problems in abstract algebra can be reformulated as structured linear approximation problems. Problems such as finding polynomials with a GCD which are nearest to given relatively prime polynomials, finding the nearest multivariate polynomial which factors, and finding the nearest polynomial which functionally decomposes, can all be cast as so-called Structured Total Least Squares (STLS) problems. More generally, given a basis of “structure” matrices X1, ...,Xm for a vector spaceV of structured matrices, and a matrix A in V, the STLS problem seeks to find the nearest B to A in V such thatB is singular. Many problems in abstract algebra can be reformulated as structured linear approxi","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125998341","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Factorization of multivariate polynomials by extended Hensel construction","authors":"D. Inaba","doi":"10.1145/1080368.1080370","DOIUrl":"https://doi.org/10.1145/1080368.1080370","url":null,"abstract":"The extended Hensel construction is a Hensel construction at an unlucky evaluation point for the generalized Hensel construction, and it allows as to avoid shifting the origin in multivariate polynomial factorization. We have implemented a multivariate factorization algorithm which is based on the extended Hensel construction, by solving a leading coefficient problem which is peculiar to our method. We describe the algorithm and present some experimental results. Experiments show that the extended Hensel construction is quite useful for factoring multivariate polynomials which cause large expression swell by shifting the origin.","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"173 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133313004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}