Factorization Of Multivariate Polynomials Research Articles

Multivariate to bivariate reduction for noncommutative polynomial factorization

Based on Bergman's theorem, we show that multivariate noncommutative polynomial factorization is deterministic polynomial-time reducible to the factorization of bivariate noncommutative polynomials. More precisely,1.Given an n-variate noncommutative polynomial f∈F〈X〉 over a field F as an arithmetic circuit, computing a complete factorization of f into irreducible factors is deterministic polynomial-time reducible to factorization of a noncommutative bivariate polynomial g∈F〈x,y〉; the reduction transforms f into a circuit for g, and given a complete factorization of g, the reduction recovers a complete factorization of f in polynomial time.The reduction works both in the white-box and the black-box setting.2.We show over the field of rationals that bivariate linear matrix factorization problem for 4×4 matrices is at least as hard as factoring square-free integers and for 3×3 matrices it is in polynomial time.

Implementation of a noise-robust quantum algorithm for multivariate polynomial factorization

We implement in an IBM quantum computer a quantum algorithm for multivariate polynomial factoring and propose a noise-avoiding protocol to distill the experimental results in the presence of noise. This algorithm uses single-qubit quantum state tomography (QST) processes to factor a specific type of multivariate polynomials. In one-to-one correspondence, it encodes each multivariate polynomial to one quantum state. While the validity of the algorithm is experimentally verified, the quality of the final results is subjected to the decoherence levels of the preparation of the quantum states. In this paper we propose a protocol to ensure the validity of factors found by our algorithm in the presence of such decoherence and noise. This method might be, in fact, part of a larger class of methods based on that same premise and useful outside the implementation of this specific algorithm. In combination with the noise robustness of the single-qubit QST, our factorization algorithm performs perfectly, even reversing the effects of weak noise, for the second- to fifth-order polynomial cases for which it has been implemented.

Discovering the Roots: Uniform Closure Results for Algebraic Classes Under Factoring

Newton iteration is an almost 350-year-old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all roots simultaneously. In this form, the process yields better circuit complexity in the case when the number of rootsris small but the multiplicities are exponentially large. Our method sets up a linear system inrunknowns and iteratively builds the roots as formal power series. For an algebraic circuit\( f(x_1,\ldots ,x_n) \)of sizes, we prove that each factor has size at most a polynomial insand the degree of the squarefree part off. Consequently, if\( f_1 \)is a\( 2^{\Omega (n)} \)-hard polynomial, then any nonzero multiple\( \prod _{i} f_i^{e_i} \)is equally hard for arbitrary positive\( e_i \)’s, assuming that\( \sum _i\deg (f_i) \)is at most\( 2^{O(n)} \).It is an old open question whether the class of poly(n) size formulas (respectively, algebraic branching programs) is closed under factoring. We show that given a polynomialfof degree\( n^{O(1)} \)and formula (respectively, algebraic branching program) size\( n^{O(\log n)} \), we can find a similar-size formula (respectively, algebraic branching program) factor in randomized poly(\( n^{\log n} \)) time. Consequently, if the determinant requires an\( n^{\Omega (\log n)} \)size formula, then the same can be said about any of its nonzero multiples.In all of our proofs, we exploit the following property of multivariate polynomial factorization. Under a random linear transformation\( \tau \), the polynomial\( f(\tau \overline{x}) \)completely factors via power series roots. Moreover, the factorization adapts well to circuit complexity analysis. Therefore, with the help of the strong mathematical characterizations and the ‘allRootsNI’ technique, we make significant progress towards the old open problems; supplementing the vast body of classical results and concepts in algebraic circuit factorization (e.g., [17,51,54,111]).

Jenks prize announcement

Pari/GP was designed for algebraic number theory. As such it does not have some standard features that other general purpose computer algebra systems have such as Gröbner bases and multivariate polynomial factorization. In this letter I say a few words about the history of Pari/GP, the impact Pari/GP has had on number theory, and some of the unique software design features of Pari/GP. I've also included a short bio from Henri, Bill and Karim.

Correction To: The Complexity of Factors of Multivariate Polynomials

Vladimir Lysikov kindly pointed out an error in the proof of Theorem 5.7. We provide here a corrected statement and its proof.

Computing the multilinear factors of lacunary polynomials without heights

We present a deterministic algorithm which computes the multilinear factors of multivariate lacunary polynomials over number fields. Its complexity is polynomial in ℓn where ℓ is the lacunary size of the input polynomial and n its number of variables, that is in particular polynomial in the logarithm of its degree. We also provide a randomized algorithm for the same problem of complexity polynomial in ℓ and n.Over other fields of characteristic zero and finite fields of large characteristic, our algorithms compute the multilinear factors having at least three monomials of multivariate polynomials. Lower bounds are provided to explain the limitations of our algorithm. As a by-product, we also design polynomial-time deterministic polynomial identity tests for families of polynomials which were not known to admit any.Our results are based on so-called Gap Theorem which reduce high-degree factorization to repeated low-degree factorizations. While previous algorithms used Gap Theorems expressed in terms of the heights of the coefficients, our Gap Theorems only depend on the exponents of the polynomials. This makes our algorithms more elementary and general, and faster in most cases.

Closure Results for Polynomial Factorization

$ \newcommand{\VNP}{\mathsf{VNP}} \newcommand{\VP}{\mathsf{VP}} \newcommand{\poly}{\mathsf{poly}} $ In a sequence of fundamental results in the 1980s, Kaltofen (SICOMP 1985, STOC'86, STOC'87, RANDOM'89) showed that factors of multivariate polynomials with small arithmetic circuits have small arithmetic circuits. In other words, the complexity class $\VP$ is closed under taking factors. A natural question in this context is to understand if other natural classes of multivariate polynomials, for instance, arithmetic formulas, algebraic branching programs, bounded-depth arithmetic circuits or the class $\VNP$, are closed under taking factors. In this paper, we show that all factors of degree at most $\log^a n$ of polynomials with $\poly(n)$-size depth-$k$ circuits have $\poly(n)$-size circuits of depth at most $O(k + a)$. This partially answers a question of Shpilka--Yehudayoff (Found. Trends in TCS, 2010) and has applications to hardness--randomness tradeoffs for bounded-depth arithmetic circuits. As direct applications of our techniques, we also obtain simple proofs of the following results. The complexity class $\VNP$ is closed under taking factors. This confirms Conjecture 2.1 in Bürgisser's monograph (2000) and improves upon a recent result of Dutta, Saxena and Sinhababu (STOC'18) who showed a quasipolynomial upper bound on the number of auxiliary variables and the complexity of the verifier circuit of factors of polynomials in $\VNP$. A factor of degree at most $d$ of a polynomial $P$ which can be computed by an arithmetic formula (or an algebraic branching program) of size $s$ has a formula (an algebraic branching program, resp.) of size at most $\poly(s, d^{\log d}, \deg(P))$. This result was first shown by Dutta et al. (STOC'18) and we obtain a slightly different proof as an easy consequence of our techniques. Our proofs rely on a combination of the ideas, based on Hensel lifting, developed in the polynomial factoring literature, and the depth-reduction results for arithmetic circuits, and hold over fields of characteristic zero or of sufficiently large characteristic. ------------- A preliminary version of this paper, titled ``Hardness vs Randomness for Bounded Depth Arithmetic Circuits,'' appeared in the Proceedings of the 33rd Computational Complexity Conference, 2018.

Quasi-tight framelets with high vanishing moments derived from arbitrary refinable functions

Construction of multivariate tight framelets is known to be a challenging problem because it is linked to the difficult problem on sum of squares of multivariate polynomials in real algebraic geometry. Multivariate dual framelets with vanishing moments generalize tight framelets and are not easy to be constructed either, since their construction is related to syzygy modules and factorization of multivariate polynomials. On the other hand, compactly supported multivariate framelets with directionality or high vanishing moments are of interest and importance in both theory and applications. In this paper we introduce the notion of a quasi-tight framelet, which is a dual framelet, but behaves almost like a tight framelet. Let ϕ∈L2(Rd) be an arbitrary compactly supported real-valued M-refinable function with a general dilation matrix M and ϕˆ(0)=1 such that its underlying real-valued low-pass filter satisfies the basic sum rule. We first constructively prove by a step-by-step algorithm that we can always easily derive from the arbitrary M-refinable function ϕ a directional compactly supported real-valued quasi-tight M-framelet in L2(Rd) associated with a directional quasi-tight M-framelet filter bank, each of whose high-pass filters has one vanishing moment and only two nonzero coefficients. If in addition all the coefficients of its low-pass filter are nonnegative, then such a quasi-tight M-framelet becomes a directional tight M-framelet in L2(Rd). Furthermore, we show by a constructive algorithm that we can always derive from the arbitrary M-refinable function ϕ a compactly supported quasi-tight M-framelet in L2(Rd) with the highest possible order of vanishing moments. We shall also present a result on quasi-tight framelets whose associated high-pass filters are purely differencing filters with the highest order of vanishing moments. Several examples will be provided to illustrate our main theoretical results and algorithms in this paper.

Factors of low individual degree polynomials

Kaltofen (Randomness in computation, vol 5, pp 375---412, 1989) proved the remarkable fact that multivariate polynomial factorization can be done efficiently, in randomized polynomial time. Still, more than twenty years after Kaltofen's work, many questions remain unanswered regarding the complexity aspects of polynomial factorization, such as the question of whether factors of polynomials efficiently computed by arithmetic formulas also have small arithmetic formulas, asked in Kopparty et al. (2014), and the question of bounding the depth of the circuits computing the factors of a polynomial. We are able to answer these questions in the affirmative for the interesting class of polynomials of bounded individual degrees, which contains polynomials such as the determinant and the permanent. We show that if $${P(x_{1},\ldots,x_{n})}$$P(x1,?,xn) is a polynomial with individual degrees bounded by r that can be computed by a formula of size s and depth d, then any factor $${f(x_{1},\ldots, x_{n})}$$f(x1,?,xn) of $${P(x_{1},\ldots,x_{n})}$$P(x1,?,xn) can be computed by a formula of size $${\textsf{poly}((rn)^{r},s)}$$poly((rn)r,s) and depth d + 5. This partially answers the question above posed in Kopparty et al. (2014), who asked if this result holds without the dependence on r. Our work generalizes the main factorization theorem from Dvir et al. (SIAM J Comput 39(4):1279---1293, 2009), who proved it for the special case when the factors are of the form $${f(x_{1}, \ldots, x_{n}) \equiv x_{n} - g(x_{1}, \ldots, x_{n-1})}$$f(x1,?,xn)?xn-g(x1,?,xn-1). Along the way, we introduce several new technical ideas that could be of independent interest when studying arithmetic circuits (or formulas).

Bounded-degree factors of lacunary multivariate polynomials

In this paper, we present a new method for computing bounded-degree factors of lacunary multivariate polynomials. In particular for polynomials over number fields, we give a new algorithm that takes as input a multivariate polynomial f in lacunary representation and a degree bound d and computes the irreducible factors of degree at most d of f in time polynomial in the lacunary size of f and in d. Our algorithm, which is valid for any field of zero characteristic, is based on a new gap theorem that enables reducing the problem to several instances of (a) the univariate case and (b) low-degree multivariate factorization.The reduction algorithms we propose are elementary in that they only manipulate the exponent vectors of the input polynomial. The proof of correctness and the complexity bounds rely on the Newton polytope of the polynomial, where the underlying valued field consists of Puiseux series in a single variable.

Extracting sparse factors from multivariate integral polynomials

In this paper we present a new algorithm for extracting sparse factors from multivariate integral polynomials. The method hinges on a new type of substitution, which reduces multivariate integral polynomials to bivariate polynomials over finite fields and keeps the sparsity of the polynomial. We retrieve the multivariate sparse factors, term by term, using discrete logarithms. We show that our method is really effective when used for factoring multivariate polynomials that have only sparse factors and when used to extract sparse factors of multivariate polynomials that may also have dense factors.

An Interpolation Procedure for List Decoding Reed–Solomon Codes Based on Generalized Key Equations

The key step of syndrome-based decoding of Reed-Solomon codes up to half the minimum distance is to solve the so-called Key Equation. List decoding algorithms, capable of decoding beyond half the minimum distance, are based on interpolation and factorization of multivariate polynomials. This article provides a link between syndrome-based decoding approaches based on Key Equations and the interpolation-based list decoding algorithms of Guruswami and Sudan for Reed-Solomon codes. The original interpolation conditions of Guruswami and Sudan for Reed-Solomon codes are reformulated in terms of a set of Key Equations. These equations provide a structured homogeneous linear system of equations of Block-Hankel form, that can be solved by an adaption of the Fundamental Iterative Algorithm. For an $(n,k)$ Reed-Solomon code, a multiplicity $s$ and a list size $\listl$, our algorithm has time complexity \ON{\listl s^4n^2}.

Computing monodromy via continuation methods on random Riemann surfaces

We consider a Riemann surface X defined by a polynomial f(x,y) of degree d, whose coefficients are chosen randomly. Hence, we can suppose that X is smooth, that the discriminant δ(x) of f has d(d−1) simple roots, Δ, and that δ(0)≠0, i.e. the corresponding fiber has d distinct points {y1,…,yd}. When we lift a loop 0∈γ⊂C−Δ by a continuation method, we get d paths in X connecting {y1,…,yd}, hence defining a permutation of that set. This is called monodromy.Here we present experimentations in Maple to get statistics on the distribution of transpositions corresponding to loops around each point of Δ. Multiplying families of “neighbor” transpositions, we construct permutations and the subgroups of the symmetric group they generate. This allows us to establish and study experimentally two conjectures on the distribution of these transpositions and on transitivity of the generated subgroups.Assuming that these two conjectures are true, we develop tools allowing fast probabilistic algorithms for absolute multivariate polynomial factorization, under the hypothesis that the factors behave like random polynomials whose coefficients follow uniform distributions.

On factoring parametric multivariate polynomials

This paper presents a new algorithm for the absolute factorization of parametric multivariate polynomials over the field of rational numbers. This algorithm decomposes the parameters space into a finite number of constructible sets. The absolutely irreducible factors of the input parametric polynomial are given uniformly in each constructible set. The algorithm is based on a parametric version of Hensel's lemma and an algorithm for quantifier elimination in the theory of algebraically closed field in order to reduce the problem of finding absolute irreducible factors to that of representing solutions of zero-dimensional parametric polynomial systems. The complexity of this algorithm is single exponential in the number n of the variables of the input polynomial, its degree d w.r.t. these variables and the number r of the parameters.

Approximate factorization of multivariate polynomials using singular value decomposition

We describe the design, implementation and experimental evaluation of new algorithms for computing the approximate factorization of multivariate polynomials with complex coefficients that contain numerical noise. Our algorithms are based on a generalization of the differential forms introduced by W. Ruppert and S. Gao to many variables, and use singular value decomposition or structured total least squares approximation and Gauss–Newton optimization to numerically compute the approximate multivariate factors. We demonstrate on a large set of benchmark polynomials that our algorithms efficiently yield approximate factorizations within the coefficient noise even when the relative error in the input is substantial (10 −3).

Improved dense multivariate polynomial factorization algorithms

We present new deterministic and probabilistic algorithms that reduce the factorization of dense polynomials from several variables to one variable. The deterministic algorithm runs in sub-quadratic time in the dense size of the input polynomial, and the probabilistic algorithm is softly optimal when the number of variables is at least three. We also investigate the reduction from several to two variables and improve the quantitative version of Bertini’s irreducibility theorem.

Factorization of multivariate positive Laurent polynomials

Recently Dritschel proved that any positive multivariate Laurent polynomial can be factorized into a sum of square magnitudes of polynomials. We first give another proof of the Dritschel theorem. Our proof is based on the univariate matrix Fejér–Riesz theorem. Then we discuss a computational method to find approximates of polynomial matrix factorization. Some numerical examples will be shown. Finally we discuss how to compute nonnegative Laurent polynomial factorizations in the multivariate setting.

Factorization of multivariate polynomials by extended Hensel construction

The extended Hensel construction is a Hensel construction at an unlucky evaluation point for the generalized Hensel construction, and it allows as to avoid shifting the origin in multivariate polynomial factorization. We have implemented a multivariate factorization algorithm which is based on the extended Hensel construction, by solving a leading coefficient problem which is peculiar to our method. We describe the algorithm and present some experimental results. Experiments show that the extended Hensel construction is quite useful for factoring multivariate polynomials which cause large expression swell by shifting the origin.

The Complexity of Factors of Multivariate Polynomials

The existence of string functions, which are not polynomial time computable, but whose graph is checkable in polynomial time, is a basic assumption in cryptography. We prove that in the framework o...

The Complexity of Factors of Multivariate Polynomials

The existence of string functions, which are not polynomial time computable, but whose graph is checkable in polynomial time, is a basic assumption in cryptography. We prove that in the framework of algebraic complexity, there are no such families of polynomial functions of polynomially bounded degree over fields of characteristic zero. The proof relies on a polynomial upper bound on the approximative complexity of a factor g of a polynomial f in terms of the (approximative) complexity of f and the degree of the factor g. This extends a result by Kaltofen (STOC 1986). The concept of approximative complexity allows to cope with the case that a factor has an exponential multiplicity, by using a perturbation argument. Our result extends to randomized (two-sided error) decision complexity.