On Efficient Noncommutative Polynomial Factorization via Higman Linearization

Multivariate to bivariate reduction for noncommutative polynomial factorization

Polynomial decomposition algorithms

This paper presents an efficient technique for synthesis and optimization of polynomials over GF(2m), where mis a non-zero positive integer. The technique is based on a graph-based decomposition and factorization of polynomials over GF(2m), followed by efficient network factorization and optimization. A technique for efficiently computing coefficients over GF(pm), where p is a prime number, is first presented. The coefficients are stored as polynomial graphs over GF(pm). The synthesis and optimization is initiated from this graph based representation. The technique has been applied to minimize multipliers over all the 51 fields in GF(2k), k = 2... 8 in 0.18 micron CMOS technology with the help of the Synopsysreg design compiler. It has also been applied to minimize combinational exponentiation circuits, and other multivariate bit- as well as word-level polynomials. The experimental results suggest that the proposed technique can reduce area, delay, and power by significant amount

This paper presents an efficient technique for synthesis and optimization of polynomials over GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ), where mis a non-zero positive integer. The technique is based on a graph-based decomposition and factorization of polynomials over GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ), followed by efficient network factorization and optimization. A technique for efficiently computing coefficients over GF(p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ), where p is a prime number, is first presented. The coefficients are stored as polynomial graphs over GF(p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ). The synthesis and optimization is initiated from this graph based representation. The technique has been applied to minimize multipliers over all the 51 fields in GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> ), k = 2... 8 in 0.18 micron CMOS technology with the help of the Synopsysreg design compiler. It has also been applied to minimize combinational exponentiation circuits, and other multivariate bit- as well as word-level polynomials. The experimental results suggest that the proposed technique can reduce area, delay, and power by significant amount

An algorithm that provides direct, efficient ND polynomial factorization is presented to solve the numerical issues that arise during the direct inversion of helium atom scattering (HAS) diffraction spectra. For an n-variate polynomial the algorithm directly deflates the polynomial to n-single variable equations by evaluating the ratio of pairs of polynomial coefficients. Error estimation of the coefficients of the 1D polynomials is then performed automatically using standard 1D search techniques. The effectiveness of the technique is demonstrated against bi- and trivariate polynomials and an approximate range of validity for error prone polynomials is demonstrated. To demonstrate the effectiveness of the technique, HAS diffraction spectra for the low coverage (2 × 1)-H/Pd(311) system have been analyzed using direct inversion and have revealed that H binds in a three-fold hollow site.

In this paper we study the complexity of factorization of polynomials in the free noncommutative ring \(\mathbb {F}\langle x_1,x_2,\ldots ,x_n \rangle \) of polynomials over the field \(\mathbb {F}\) and noncommuting variables \(x_1,x_2,\ldots ,x_n\). Our main results are the following: Although \(\mathbb {F}\langle x_1,\ldots ,x_n \rangle \) is not a unique factorization ring, we note that variable-disjoint factorization in \(\mathbb {F}\langle x_1,\ldots ,x_n \rangle \) has the uniqueness property. Furthermore, we prove that computing the variable-disjoint factorization is polynomial-time equivalent to Polynomial Identity Testing (both when the input polynomial is given by an arithmetic circuit or an algebraic branching program). We also show that variable-disjoint factorization in the black-box setting can be efficiently computed (where the factors computed will be also given by black-boxes, analogous to the work [12] in the commutative setting). As a consequence of the previous result we show that homogeneous noncommutative polynomials and multilinear noncommutative polynomials have unique factorizations in the usual sense, which can be efficiently computed. Finally, we discuss a polynomial decomposition problem in \(\mathbb {F}\langle x_1,\ldots ,x_n \rangle \) which is a natural generalization of homogeneous polynomial factorization and prove some complexity bounds for it. KeywordsHomogeneous PolynomialUnique FactorizationArithmetic CircuitPolynomial FactorizationMultilinear PolynomialThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

On the complexity of noncommutative polynomial factorization

We present the architectures for bivariate polynomial interpolation and factorization; the two main steps in algebraic soft-decision decoding of Reed-Solomon codes. We present an efficient formulation of the interpolation algorithm in which dependencies among the discrepancy coefficient computations are utilized to reduce interpolation complexity. Interpolation and factorization complexity is also reduced by using an FFT-like formulation for univariate polynomial translation. The modifications required to incorporate the recently proposed algorithm level modifications for efficient interpolation and factorization are also presented. We determine the latency and hardware requirements for soft-decoding a [255,239] Reed-Solomon code using the proposed architectures.

We study certain quadratic and quadratic linear algebras related to factorizations of noncommutative polynomials and differential polynomials. Such algebras possess a natural derivation and give us a new understanding of the nature of noncommutative symmetric functions.

On the factorization of non-commutative polynomials (in free associative algebras)

Geometry of free loci and factorization of noncommutative polynomials

A rational function belongs to the Hardy space, H 2 H^2 , of square-summable power series if and only if it is bounded in the complex unit disk. Any such rational function is necessarily analytic in a disk of radius greater than one. The inner-outer factorization of a rational function r ∈ H 2 \mathfrak {r} \in H^2 is particularly simple: The inner factor of r \mathfrak {r} is a (finite) Blaschke product and (hence) both the inner and outer factors are again rational. We extend these and other basic facts on rational functions in H 2 H^2 to the full Fock space over C d \mathbb {C} ^d , identified as the non-commutative (NC) Hardy space of square-summable power series in several NC variables. In particular, we characterize when an NC rational function belongs to the Fock space, we prove analogues of classical results for inner-outer factorizations of NC rational functions and NC polynomials, and we obtain spectral results for NC rational multipliers.

Consider a polynomial with parametric coefficients. We show that the variety of parameters can be represented as a union of strata. For values of the parameters from each stratum, the decomposition of this polynomial into absolutely irreducible factors is given by algebraic formulas depending only on the stratum. Each stratum is a quasiprojective algebraic variety. This variety and the corresponding output are given by polynomials of degrees at most D with D = d′d O(1) where d′, d are bounds on the degrees of the input polynomials. The number of strata is polynomial in the size of the input data. Thus, here we avoid double exponential upper bounds for the degrees and solve a long-standing problem.

Improved Techniques for Factoring Univariate Polynomials

Polynomial Factorization: Sharp Bounds, Efficient Algorithms