We study the approximate GCD of two univariate polynomials given with limited accuracy or, equivalently, the exact GCD of the perturbed polynomials within some prescribed tolerance. A perturbed polynomial is regarded as a family of polynomials in a classification space, which leads to an accurate analysis of the computation. Considering only the Sylvester matrix singular values, as is frequently suggested in the literature, does not suffice to solve the problem completely, even when the extended euclidean algorithm is also used. We provide a counterexample that illustrates this claim and indicates the problem's hardness. SVD computations on subresultant matrices lead to upper bounds on the degree of the approximate GCD. Further use of the subresultant matrices singular values yields an approximate syzygy of the given polynomials, which is used to establish a gap theorem on certain singular values that certifies the maximum-degree approximate GCD. This approach leads directly to an algorithm for computing the approximate GCD polynomial. Lastly, we suggest the use of weighted norms in order to sharpen the theorem's conditions in a more intrinsic context.

Although a polynomial time algorithm exists, the most commonly used algorithm for factoring a univariate polynomial f with integer coefficients is the Berlekamp-Zassenhaus algorithm which has a complexity that depends exponentially on n where n is the number of modular factors of f . This exponential time complexity is due to a combinatorial problem; the problem of choosing the right subset of these n factors. In this paper we reduce this combinatorial problem to a knapsack problem of a kind that can be solved with polynomial time algorithms such LLL or PSLQ. The result is a practical algorithm that can factor large polynomials even when n is large as well. 1 Introduction Let f be a polynomial of degree N with integer coefficients, f = N X i=0 a i x i where a i 2 ZZ. Assume that f is monic (i.e. aN = 1) and that f is square-free (no multiple roots), so the gcd of f and f 0 equals 1. Let p be a prime number and let F p = ZZ=(p) be the field with p elements. Let ZZ p ...

To approximate all roots (zeros) of a univariate polynomial, we develop two effective algorithms and combine them in a single recursive process. One algorithm computes a basic well isolated zero-free annulus on the complex plane, whereas another algorithm numerically splits the input polynomial of the n-th degree into two factors balanced in the degrees and with the zero sets separated by the basic annulus. Recursive combination of the two algorithms leads to recursive computation of the complete numerical factorization of a polynomial into the product of linear factors and further to the approximation of the roots. The new rootfinder incorporates the earlier techniques of Schönhage and Kirrinnis and our old and new techniques and yields nearly optimal (up to polylogarithmic factors) arithmetic and Boolean cost estimates for the complexity of both complete factorization and rootfinding. The improvement over our previous record Boolean complexity estimates is by roughly the factor of n for complete factorization and also for the approximation of well-conditioned (well isolated) roots, whereas the same algorithm is also optimal (under both arithmetic and Boolean models of computing) for the worst case input polynomial, where the roots can be ill-conditioned, forming clusters. (The worst case bounds are supported by our previous algorithms as well.) All our algorithms allow processor efficient acceleration to achieve solution in polylogarithmic parallel time. Keywords Padé approximation, Graeffe’s lifting, univariate polynomials, rootfinding, numerical polynomial factorization, geometry of polynomial zeros, computational complexity

Abstract. Van Hoeij’s algorithm for factoring univariate polynomials over the rational integers rests on the same principle as Berlekamp-Zassenhaus, but uses lattice basis reduction to improve drastically on the recombination phase. His ideas give rise to a collection of algorithms, differing greatly in their efficiency. We present two deterministic variants, one of which achieves excellent overall performance. We then generalize these ideas to factor polynomials over

At Eurocrypt ’96, Coppersmith proposed an algorithm for finding small roots of bivariate integer polynomial equations, based on lattice reduction techniques. But the approach is difficult to understand. In this paper, we present a much simpler algorithm for solving the same problem. Our simplification is analogous to the simplification brought by Howgrave-Graham to Coppersmith’s algorithm for finding small roots of univariate modular polynomial equations. As an application, we illustrate the new algorithm with the problem of finding the factors of n = pq if we are given the high order 1/4log 2 n bits of p.

The known record complexity estimates for approximating polynomial zeros rely on geometric constructions on the complex plane, which achieve initial approximation to the zeros and/or their clusters as well as their isolation from each other, and on the subsequent fast analytic refinement of the initial approximations. We modify Weyl's classical geometric construction for approximating all the n polynomial zeros in order to more rapidly achieve their strong isolation. For approximating the isolated zeros or clusters of zeros, we propose a new extension of Newton's iteration to yield quadratic global convergence (right from the start), under substantially weaker requirements to their initial isolation than one needs in the known algorithms.

A new algorithm is presented for nding the Frobenius rational form F 2 Z nn of any A 2 Z nn which requires an expected O(n 4 (log n+log kAk)+n 3 (log n+log kAk) 2 ) word operations using standard integer and matrix arithmetic. This improves substantially on the fastest previously known algorithms. The algorithm is probabilistic of the Las Vegas type: it assumes a source of random bits but always produces the correct answer. Las Vegas algorithms are also presented for computing a transformation matrix to the Frobenius form, and for computing the rational Jordan form of an integer matrix. 1 Introduction In this paper we present new algorithms for computing exactly the Frobenius and rational Jordan normal forms of an integer matrix which are substantially faster than those previously known. We show that the Frobenius form F 2 Z nn of any A 2 Z nn can be computed with an expected number of O(n 4 (log n + log kAk) + n 3 (log n + log kAk) 2 ) word operations using s...

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A Monte Carlo type probabilistic algorithm is presented for finding the Frobenius rational form F 2 Z n\Thetan of any A 2 Z n\Thetan which requires an expected number of O(n 4 (log n + kAk) 2 ) bit operations using standard integer and matrix arithmetic (where kAk is the largest absolute value of any entry of A). This improves dramatically on the fastest previously known algorithm, which requires O(n 6 log kAk) bit operations using fast integer arithmetic. We also give a Las Vegas type probabilistic algorithm which finds the Frobenius form F and a transition matrix U 2 Q n\Thetan such that U \Gamma1 AU = F and requires an expected number of O(n 5 (log n + log kAk) 5=2 bit operations. Finally, a Las Vegas algorithm for computing the rational Jordan form of an integer matrix is shown, which requires about the same number of bit operations as our algorithm to find the Frobenius form, plus the time required to factor the characteristic polynomial of that matrix. 1. I...