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Algorithms are developed that adopt a novel implicit representation for multivariate polynomials and rational functions with rational coefficients, that of black boxes for their evaluation. It is shown that within this evaluation-box representation, the polynomial greatest common divisor and factorization problems as well as the problem of extracti...
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... discussing our results, let us mention a linear algebra setting in which black box representation also can be successfully applied. We are given the coefficient matrix A of a linear system in terms of a black box that will multiply this matrix with a chosen vector y (see Figure 3). For simplicity, assume that A ∈ K n×n is non-singular. ...
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... The algorithm in [14] is the first polynomial algorithm in the input and output sizes and the degree . In 1988 and 1990, Kaltofen and Trager [16,17] gave factorization algorithms for polynomials given by black boxes. In 2005, Kaltofen and Koiran [20] proved that bivariate irreducibility over integers is co-NP hard via randomized reductions. ...
... The algorithm in [14] outputs an SLP as the factors, whose complexity is polynomial in the input size and the degree; and when outputting sparse polynomials, the algorithm has the complexity given in Table 2. The algorithms in [16,17] are the construction of an evaluation program for all irreducible factors of a black box polynomial, and then recovered all the factors by evaluation-interpolation scheme. ...
... Given on input a bound on the number of terms of the irreducible factors, both algorithms in [16,17] and [14] output all sparse irreducible factors with ≤ terms in random polynomial time. ...
... See [11] for more details. ...
... We will assume that a circuit from class C computing the input polynomial f has a sum gate at the output. Otherwise, we can apply the factorization algorithm in [22] to gain blackbox access to all the irreducible factors of f , thereby reducing the problem to a potentially simpler class of circuits at the cost of making the reconstruction algorithm randomized. Thus, depth two, depth three and depth four circuits would mean ΣΠ, ΣΠΣ and ΣΠΣΠ circuits respectively. ...
... It follows that g = IMM w ,d (A x + b ), where A ∈ GL(n) is the matrix B · C restricted to the first n columns. 22 Which follows easily from Claim 26. C C C 2 0 1 7 ...
... Assuming irreducibility of input f in Algorithm 2: The idea is to construct blackbox access to the irreducible factors of f using the efficient randomized polynomial factorization algorithm in [22], and compute full rank ABP for each of these irreducible factors. The ABPs are then connected 'in series' to form a full rank ABP for f . ...
An algebraic branching program (ABP) A can be modelled as a product expression X1amp;middot; X2… Xd, where X1 and Xd are 1 × w and w × 1 matrices, respectively, and every other Xk is a w × w matrix; the entries of these matrices are linear forms in m variables over a field F (which we assume to be either Q or a field of characteristic poly(m)). The polynomial computed by A is the entry of the 1 × 1 matrix obtained from the product ∏k=1d Xk. We say A is a full rank ABP if the w²(d − 2) + 2w linear forms occurring in the matrices X1, X2, …, Xd are F-linearly independent. Our main result is a randomized reconstruction algorithm for full rank ABPs: Given blackbox access to an m-variate polynomial f of degree at most m, the algorithm outputs a full rank ABP computing f if such an ABP exists, or outputs “no full rank ABP exists” (with high probability). The running time of the algorithm is polynomial in m and β, where β is the bit length of the coefficients of f. The algorithm works even if Xk is a wk−1 × wk matrix (with w0 = wd = 1), and w = (w1, …, wd − 1) is unknown. The result is obtained by designing a randomized polynomial time equivalence test for the family of iterated matrix multiplication polynomial IMMw, d, the (1, 1)-th entry of a product of d rectangular symbolic matrices whose dimensions are according to w ∈ Nd−1. At its core, the algorithm exploits a connection between the irreducible invariant subspaces of the Lie algebra of the group of symmetries of a polynomial f that is equivalent to IMMw, d and the “layer spaces” of a full rank ABP computing f. This connection also helps determine the group of symmetries of IMMw, d and show that IMMw, d is characterized by its group of symmetries.
... Multivariate sparse rational function interpolation with algorithms that are polynomial in the degrees and number of variables and terms in g and h are presented in [13] [18] [21]. An important ingredient are Zippel's or Ben-Or and Tiwari's sparse multivariate polynomial interpolation algorithms (see, e.g., [17] and the references there). ...
We present a method for interpolating a supersparse blackbox rational function with rational coefficients, for example, a ratio of binomials or trinomials with very high degree. We input a blackbox rational function, as well as an upper bound on the number of non-zero terms and an upper bound on the degree. The result is found by interpolating the rational function modulo a small prime p, and then applying an effective version of Dirichlet's Theorem on primes in an arithmetic progression progressively lift the result to larger primes. Eventually we reach a prime number that is larger than the inputted degree bound and we can recover the original function exactly. In a variant, the initial prime p is large, but the exponents of the terms are known modulo larger and larger factors of p-1. The algorithm, as presented, is conjectured to be polylogarithmic in the degree, but exponential in the number of terms. Therefore, it is very effective for rational functions with a small number of non-zero terms, such as the ratio of binomials, but it quickly becomes ineffective for a high number of terms. The algorithm is oblivious to whether the numerator and denominator have a common factor. The algorithm will recover the sparse form of the rational function, rather than the reduced form, which could be dense. We have experimentally tested the algorithm in the case of under 10 terms in numerator and denominator combined and observed its conjectured high efficiency.
... Note that, due to the restrictions on the degrees of g and h, the linear system given by (2) has 2mn + m + n unknowns and at most 4mn equations. Following Gao [6], it can be solved by the black box approach of Kaltofen and Trager [13]. This leads to a basis B of the vector space G with O(dmn) matrix-vector products, each of which can be computed by three multiplications of polynomials with bidegree at most (m, n), hence using O(mn log 2 (mn)) field operations. ...
Shuhong Gao (2003) [6] has proposed an efficient algorithm to factor a bivariate polynomial f over a field F. This algorithm is based on a simple partial differential equation and depends on a crucial fact: the dimension of the polynomial solution space G associated with this differential equation is equal to the number r of absolutely irreducible factors of f. However, this holds only when the characteristic of F is either zero or sufficiently large in terms of the degree of f. In this paper we characterize a vector subspace of G for which the dimension is r, regardless of the characteristic of F, and the properties of Gao’s construction hold. Moreover, we identify a second vector subspace of G that leads to an analogous theory for the rational factorization of f.
... The black box model for polynomials [40] and the parallel reconstruction algorithm via sparse interpolation seemed much more practicable—embarrassingly parallel . Thus in 1990 we began developing a run-time support tool, DSC (Distributed Symbolic Computation) [10] [8]. ...
A second wave of parallel and distributed computing research is rolling in. Today's multicore/multiprocessor computers facilitate everyone's parallel execution. In the mid 1990s, manufactures of expensive main-frame parallel computers faltered and computer science focused on the Internet and the computing grid. After a ten year hiatus, the Parallel Symbolic Computation Conference (PASCO) is awakening with new vigor. I shall look back on the highlights of my own research on theoretical and practical aspects of parallel and distributed symbolic computation, and forward to what is to come by example of several current projects. An important technique in symbolic computation is the evaluation/interpolation paradigm, and multivariate sparse polynomial parallel interpolation constitutes a keystone operation, for which we present a new algorithm. Several embarrassingly parallel searches for special polynomials and exact sum-of-squares certificates have exposed issues in even today's multiprocessor architectures. Solutions are in both software and hardware. Finally, we propose the paradigm of interactive symbolic supercomputing, a symbolic computation environment analog of the STAR-P Matlab platform.
... , e ct ∈ C we compute to also be relatively close to their values in the exact computation. Black-box polynomials appear naturally in applications such as polynomial systems [5] and the manipulation of sparse polynomials (e.g., factoring polynomials [16] [6]). Sparsity with respect to the power (or other) basis is also playing an ever more important role in computer algebra. ...
... Proof and algorithm. The primitive-element algorithm of Canny [Can88] (or the black-box method of Kaltofen and Villard [KT88]) reduces factoring to the computation and manipulation of 2n þ 1 univariate polynomials in u 0 ; denoted by R 0 ; R þ i ; R À i ; i ¼ 1; y; n: For details, see the proof of Lemma 2.2 in [Can88]. Each new polynomial is defined by specializing the variables u 1 ; y; u n to randomly selected constants, hence yielding univariate polynomials in u 0 : These polynomials have degree V 0 : Their output-sensitive coefficient length is in OðlogjjRðu 0 ÞjjÞ: Another bound on this length is V 0 times the length of the coefficients in RðuÞ; hence OðV 0 DcÞ: ...
Our first contribution is a substantial acceleration of randomized computation of scalar, univariate, and multivariate matrix determinants, in terms of the output-sensitive bit operation complexity bounds, including computation modulo a product of random primes from a fixed range. This acceleration is dramatic in a critical application, namely solving polynomial systems and related studies, via computing the resultant. This is achieved by combining our techniques with the primitive-element method, which leads to an effective implicit representation of the roots. We systematically examine quotient formulae of Sylvester-type resultant matrices, including matrix polynomials and the u-resultant. We reduce the known bit operation complexity bounds by almost an order of magnitude, in terms of the resultant matrix dimension. Our theoretical and practical improvements cover the highly important cases of sparse and degenerate systems.
... There is an extensive literature on this problem — we refer the reader to the references in [6]. Most of these papers deal with dense polynomials, two notable exceptions being [7] [9]. These two papers reduce sparse polynomials with more than two variables to bivariate or univariate polynomials which are then treated as dense polynomials. ...
We introduce a new approach to multivariate polynomial factorisation which incorporates ideas from polyhedral geometry, and generalises Hensel lifting. Our main contribution is to present an algorithm for factoring bivariate polynomials which is able to exploit to some extent the sparsity of polynomials. We give details of an implementation which we used to factor randomly chosen sparse and composite polynomials of high degree over the binary field.
... Proof and algorithm. The primitive-element algorithm of Canny [Can88] (or the black-box method of Kaltofen and Trager [KT88]) reduces factoring to the computation and manipulation of 2n þ 1 univariate polynomials in u 0 ; denoted by R 0 ; R þ i ; R À i ; i ¼ 1; y; n: For details, see the proof of Lemma 2.2 in [Can88]. Each new polynomial is defined by specializing the variables u 1 ; y; u n to randomly selected constants, hence yielding univariate polynomials in u 0 : These polynomials have degree V 0 : Their output-sensitive coefficient length is in OðlogjjRðu 0 ÞjjÞ: Another bound on this length is V 0 times the length of the coefficients in RðuÞ; hence OðV 0 DcÞ: ...
Our first contribution is a substantial acceleration of randomized computation of scalar, univariate, and multivariate matrix determinants, in terms of the output-sensitive bit operation complexity bounds, including computation modulo a product of random primes from a fixed range. This acceleration is dramatic in a critical application, namely solving polynomial systems and related studies, via computing the resultant. This is achieved by combining our techniques with the primitive-element method, which leads to an effective implicit representation of the roots. We systematically examine quotient formulae of Sylvester-type resultant matrices, including matrix polynomials and the u-resultant. We reduce the known bit operation complexity bounds by almost an order of magnitude, in terms of the resultant matrix dimension. Our theoretical and practical improvements cover the highly important cases of sparse and degenerate systems.
