Polynomial factorization: a success story

• Authors: Erich Kaltofen
• Venue: ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
• Year: 2003

The problem of factoring a polynomial in a single or severalvariables over a finite field, the rational numbers or the complexnumbers is one of the success stories in the discipline of symboliccomputation. In the early 1960s implementors investigated theconstructive methods known from classical algebra books, but--withthe exception of Gauss's distinct degree factorizationalgorithm--found the algorithms quite inefficient in practice [16].The contributions in algorithmic techniques that have been madeover the next 40 years are truly a hallmark of symbolic computationresearch.The early pioneers, Berlekamp, Musser, Wang, Weinberger,Zassenhaus and others applied new ideas like randomization, thateven before the now famous algorithms for primality testing byRabin and Strassen, and like generic programming with coefficientdomains as abstract data classes, and they introduced the powerfulHensel lifting lemma to computer algebra. We note that whilede-randomization for integer primality testing has beenaccomplished recently [1], the same remains open for the problem ofcomputing a root of a polynomial modulo a large prime [12, ResearchProblem 14.48].Polynomial-time complexity for rational coefficients wasestablished in the early 1980s by the now-famous lattice basisreduction algorithm of A. Lenstra, H. W. Lenstra, Jr., and L.Lovász. The case of many variables first became anapplication of the DeMillo and Lipton/Schwartz/Zippel lemma [30]and then triggered a fundamental generalization from the standardsparse (distributed) representation of polynomials to the one bystraight line and black box programs [11, 17, 19]. Effectiveversions of the Hilbert irreducibility theorem are needed for theprobabilistic analysis, which serendipitously later have alsoplayed a role in the PCP characterization of NP[2]. Unlike many other problems in commutative algebra andalgebraic geometry, such as algebraic system solving, thepolynomial factoring problem is of probabilistic polynomial-timecomplexity in the number of variables.Complex coefficients in multivariate factors can be representedeither by exact algebraic numbers or by imprecise floating pointnumbers. The latter formulation is a cornerstone in the newcomputer algebra subject of SNAP (Symbolic-Numeric Algorthms forPolynomials) (see, e.g., [4]). The approaches for both exact andimprecise coefficients are manifold, including Ruppert's partialdifferential equations [26, 27, 6, 10] and Gao's and Lauder'sfar-reaching generalization of Eisenstein's criterion in themultivariate case to Newton polytope decomposition [8, 9]. Thecurrently best algorithms were all discovered recently within thepast ten years.The baby steps/giant steps technique and fast distinct and equaldegree factorization implementations have, at last, yielded in themid 1990s theoretical and practical improvements over the originalunivariate Berlekamp algorithm for coefficients in finite fields[13, 29, 18, 3]. The average time analysis for selected algorithmsis also completed [5]. For bivariate polynomials over finitefields, surprisingly Gröbner basis techniques are useful inpractice [23].New polynomial-time complexity results are the computation oflow degree factors of very high degree sparse (lacunary)polynomials by H. W. Lenstra, Jr. [20, 21], and the deterministicdistinct degree factorization for multivariate polynomials overlarge finite fields [7]. However, many problems with high degreepolynomials over large finite fields in sparse or straight lineprogram representations, such as computing a root modulo a largeprime, are not known to be in random polynomial time or NP-hard(cf. [24, 25, 15]).Finally, in 2000 Mark van Hoeij [14] reintroduced lattice basisreduction, now in the Berlekamp-Zassenhaus algorithm, to conquerthe hard-to-factor Swinnerton-Dyer polynomials in practice. Sasakiin 1993 had already hinted of the used approach [28].In my talk I will discuss a selection of the highlights, stateremaining open problems, and give some applications including anunusual one due to Moni Naor [22].