{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article1518","name":"Two Families of Algorithms for Symbolic Polynomials","abstract":"We wish to work with polynomials where the exponents are not known \r\nin advance, such as $x^{2n} - 1$. There are various operations we will\r\nwant to be able to do, such as squaring the value to get $x^{4n}-2x^{2n}+1$,\r\nor differentiating it to get $2nx^{2n-1}$. Expressions of this sort\r\narise frequently in practice, for example in the analysis of algorithms,\r\nand it is very difficult to work with them effectively in current computer\r\nalgebra systems.\r\n\r\nWe consider the case where multivariate polynomials can have exponents\r\nwhich are themselves integer-valued multivariate polynomials, and we present\r\nalgorithms to compute their GCD and factorization. The algorithms fall into\r\ntwo families: algebraic extension methods and interpolation methods.\r\nThe first family of algorithms uses the algebraic independence of $x$, $x^n$,\r\n$x^{n^2}$, $x^{nm}, etc, to solve related problems with more indeterminates. \r\nSome subtlety is needed to avoid problems with fixed divisors of the exponent\r\npolynomials. The second family of algorithms uses evaluation and interpolation\r\nof the exponent polynomials. While these methods can run into unlucky\r\nevaluation points, in many cases they can be more appealing. Additionally,\r\nwe also treat the case of symbolic exponents on rational coefficients\r\n(e.g. $4^{n^2+n}-81$) and show how to avoid integer factorization.","keywords":["Computer algebra","symbolic computation","factorization","gcd","symbolic exponents"],"author":{"@type":"Person","name":"Watt, Stephen M.","givenName":"Stephen M.","familyName":"Watt"},"position":15,"pageStart":1,"pageEnd":20,"dateCreated":"2006-11-07","datePublished":"2006-11-07","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Watt, Stephen M.","givenName":"Stephen M.","familyName":"Watt"},"copyrightYear":"2006","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.06271.15","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume615","volumeNumber":6271,"name":"Dagstuhl Seminar Proceedings, Volume 6271","dateCreated":"2006-10-25","datePublished":"2006-10-25","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article1518","isPartOf":{"@type":"Periodical","@id":"#series119","name":"Dagstuhl Seminar Proceedings","issn":"1862-4405","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume615"}}}