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12
Factoring Multivariate Polynomials via Partial Differential Equations
 Math. Comput
, 2000
"... A new method is presented for factorization of bivariate polynomials over any field of characteristic zero or of relatively large characteristic. It is based on a simple partial differential equation that gives a system of linear equations. Like Berlekamp's and Niederreiter's algorithms fo ..."
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Cited by 50 (9 self)
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A new method is presented for factorization of bivariate polynomials over any field of characteristic zero or of relatively large characteristic. It is based on a simple partial differential equation that gives a system of linear equations. Like Berlekamp's and Niederreiter's algorithms for factoring univariate polynomials, the dimension of the solution space of the linear system is equal to the number of absolutely irreducible factors of the polynomial to be factored and any basis for the solution space gives a complete factorization by computing gcd's and by factoring univariate polynomials over the ground field. The new method finds absolute and rational factorizations simultaneously and is easy to implement for finite fields, local fields, number fields, and the complex number field. The theory of the new method allows an effective Hilbert irreducibility theorem, thus an efficient reduction of polynomials from multivariate to bivariate.
Algorithms in algebraic number theory
 Bull. Amer. Math. Soc
, 1992
"... Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to ..."
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Cited by 42 (4 self)
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Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion is concentrated of three topics: the determination of Galois groups, the determination of the ring of integers of an algebraic number field, and the computation of the group of units and the class group of that ring of integers. 1.
Factoring polynomials over padic fields
 ANTS IV, VOLUME 1838 OF LNCS
, 2000
"... We give an efficient algorithm for factoring polynomials over finite algebraic extensions of the padic numbers. This algorithm uses ideas of Chistov’s random polynomialtime algorithm, and is suitable for practical implementation. ..."
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Cited by 3 (0 self)
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We give an efficient algorithm for factoring polynomials over finite algebraic extensions of the padic numbers. This algorithm uses ideas of Chistov’s random polynomialtime algorithm, and is suitable for practical implementation.
Factoring Modular Polynomials
 In International Symposium on Symbolic and Algebraic Computation
, 1996
"... This paper gives an algorithm to factor a polynomial f (in one variable) over rings like Z=rZ for r 2 Z or F q [y]=rF q [y] for r 2 F q [y]. The Chinese Remainder Theorem reduces our problem to the case where r is a prime power. Then factorization is not unique, but if r does not divide the discrimi ..."
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Cited by 1 (0 self)
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This paper gives an algorithm to factor a polynomial f (in one variable) over rings like Z=rZ for r 2 Z or F q [y]=rF q [y] for r 2 F q [y]. The Chinese Remainder Theorem reduces our problem to the case where r is a prime power. Then factorization is not unique, but if r does not divide the discriminant of f , our (probabilistic) algorithm produces a description of all (possibly exponentially many) factorizations into irreducible factors in polynomial time. If r divides the discriminant, we only know how to factor by exhaustive search, in exponential time.
Faster padic Feasibility for Certain Multivariate Sparse Polynomials
"... Wepresentalgorithmsrevealingnewfamiliesofpolynomialsadmittingsubexponentialdetection of padic rational roots, relative to the sparse encoding. For instance, we prove NPcompleteness for the case of honest nvariate (n+1)nomials and, for certain special cases with p exceeding the Newton polytope v ..."
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Wepresentalgorithmsrevealingnewfamiliesofpolynomialsadmittingsubexponentialdetection of padic rational roots, relative to the sparse encoding. For instance, we prove NPcompleteness for the case of honest nvariate (n+1)nomials and, for certain special cases with p exceeding the Newton polytope volume, constanttime complexity. Furthermore, using the theory of linear forms in padic logarithms, we prove that the case of trinomials in one variable can be done in NP. The best previous complexity upper bounds for all these problems were EXPTIME or worse.Finally,weprovethatdetectingpadicrationalrootsforsparsepolynomialsinonevariable is NPhard with respect to randomized reductions. The last proof makes use of an efficient construction of primes in certain arithmetic progressions. The smallest n where detecting padic rational roots for nvariate sparse polynomials is NPhard appears to have been unknown.
Algorithmic Arithmetic Fewnomial Theory I: One Variable (Extended Abstract)
, 2007
"... We show that deciding whether a sparse polynomial in one variable has a root in Fp (for p prime) is NPhard with respect to BPP reductions. As a consequence, we answer open questions on the factorization of sparse polynomials posed by Karpinski and Shparlinski, and Cox. We also derive analogous resu ..."
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We show that deciding whether a sparse polynomial in one variable has a root in Fp (for p prime) is NPhard with respect to BPP reductions. As a consequence, we answer open questions on the factorization of sparse polynomials posed by Karpinski and Shparlinski, and Cox. We also derive analogous results for detecting padic rational roots, thus paralleling a recent complexity phase transition over the real numbers. A related new result is that detecting padic rational roots for a sparse polynomial in one variable is in NP for most inputs. Along the way, we also develop an efficient method for generating random primes in certain arithmetic progressions In the sequel to this paper, we extend our complexity results to systems of multivariate polynomials.
Summary
"... Variable selection and transformation selection are two commonly encountered problems in the linear model. It is often of interest to combine these two procedures in an analysis. Due to recent developments in computing technology, such a procedure is now feasible. In this paper, we propose two varia ..."
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Variable selection and transformation selection are two commonly encountered problems in the linear model. It is often of interest to combine these two procedures in an analysis. Due to recent developments in computing technology, such a procedure is now feasible. In this paper, we propose two variable and transformation selection procedures on the predictor variables in the linear model. The rst procedure is a simultaneous variable and transformation selection procedure. For data sets with many predictors, a backward elimination procedure for variables and transformations is also presented. The procedures are based on Bayesian model selection criteria introduced by Ibrahim and Laud (1994) and Laud and Ibrahim (1995). Several examples are given to illustrate the methodology.
Factoring Polynomials Modulo Composites
, 1997
"... This paper characterizes all the factorizations of a polynomial with coefficients in the ring Z n where n is a composite number. We give algorithms to compute such factorizations along with algebraic classifications. Contents 1 Introduction 3 1.1 Circuit complexity theory . . . . . . . . . . . . ..."
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This paper characterizes all the factorizations of a polynomial with coefficients in the ring Z n where n is a composite number. We give algorithms to compute such factorizations along with algebraic classifications. Contents 1 Introduction 3 1.1 Circuit complexity theory . . . . . . . . . . . . . . . . . . . . . . 3 2 Some Important Tools in Z n [x] 4 2.1 The Z n [x] phenomena . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . 5 2.3 Irreducibility criteria in Z p k [x] . . . . . . . . . . . . . . . . . . . 7 2.4 Hensel's Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.5 A naive approach to factoring . . . . . . . . . . . . . . . . . . . . 11 3 The Case of Small Discriminants 12 3.1 The padic numbers . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Resultants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 The correspondence to factoring over the padics . . . . ....
A Number Theoretic Interpolation Between Quantum and Classical Complexity Classes
, 2008
"... We reveal a natural algebraic problem whose complexity appears to interpolate between the wellknown complexity classes BQP and NP: ⋆ Decide whether a univariate polynomial with exactly m monomial terms has a padic rational root. In particular, we show that while (⋆) is doable in quantum randomized ..."
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We reveal a natural algebraic problem whose complexity appears to interpolate between the wellknown complexity classes BQP and NP: ⋆ Decide whether a univariate polynomial with exactly m monomial terms has a padic rational root. In particular, we show that while (⋆) is doable in quantum randomized polynomial time when m = 2 (and no classical randomized polynomial time algorithm is known), (⋆) is nearly NPhard for general m: Under a plausible hypothesis involving primes in arithmetic progression (implied by the Generalized Riemann Hypothesis for certain cyclotomic fields), a randomized polynomial time algorithm for (⋆) would imply the widely disbelieved inclusion NP⊆BPP. This type of quantum/classical interpolation phenomenon appears to new. 1 Introduction and Main Results Thanks to quantum computation, we now have exponential speedups for important practical problems such as Integer Factoring and Discrete Logarithm [Sho97]. However, a fundamental
Computing All Factorizations in Zn[x]
, 2001
"... We present a new algorithm for determining all factorizations of a polynomial f in the domain Z N [x], a nonunique factorization domain, given in terms of parameters. From the prime factorization of N , the problem is reduced to factorization in Z p k[x] where p is a prime and k 1. If p k does n ..."
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We present a new algorithm for determining all factorizations of a polynomial f in the domain Z N [x], a nonunique factorization domain, given in terms of parameters. From the prime factorization of N , the problem is reduced to factorization in Z p k[x] where p is a prime and k 1. If p k does not divide the discriminant of f and one factorization is given, our algorithm determines all factorizations with complexity O(n 3 M(k log p)) where n denotes the degree of the input polynomial and M(t) denotes the complexity of multiplication of two tbit numbers. Our algorithm improves on the method of von zur Gathen and Hartlieb, which has complexity O(n 7 k(k log p + log n) 2 ). The improvement is achieved by processing all factors at the same time instead of one at a time and by computing the kernels and determinants of matrices over Z p k in an efficient manner. Categories and Subject Descriptors I.1.2 [Symbolic and Algebraic Manipulation]: Algorithms