Factorization Of Polynomials Research Articles

Complexity of the critical node problem over trees

In this paper we deal with the critical node problem (CNP), i.e., the problem of searching for a given number K of nodes in a graph G, whose removal minimizes the (weighted or unweighted) number of connections between pairs of nodes in the residual graph. In particular, we study the case where the physical network represented by graph G has a hierarchical organization, so that G is a tree. The NP‐completeness of this problem for general graphs has been already established (Arulselvan et al.). We study the subclass of CNP over trees, generalizing the objective function and constraints to take into account general nonnegative “costs” of node connections and “weights” for the nodes that are to be removed. We prove that CNP over trees is still NP‐complete when general connection costs are specified, while the cases where all connections have unit cost are solvable in polynomial time by dynamic programming approaches. For the case with nonnegative connection costs and unit node weights we propose an enumeration scheme whose time complexity is within a polynomial factor from O(1.618034n), where n is the number of nodes of the tree. Results from computational experiments are reported for all the proposed algorithms.

Causal–anticausal decomposition of speech using complex cepstrum for glottal source estimation

Complex cepstrum is known in the literature for linearly separating causal and anticausal components. Relying on advances achieved by the Zeros of the Z-Transform (ZZT) technique, we here investigate the possibility of using complex cepstrum for glottal flow estimation on a large-scale database. Via a systematic study of the windowing effects on the deconvolution quality, we show that the complex cepstrum causal–anticausal decomposition can be effectively used for glottal flow estimation when specific windowing criteria are met. It is also shown that this complex cepstral decomposition gives similar glottal estimates as obtained with the ZZT method. However, as complex cepstrum uses FFT operations instead of requiring the factoring of high-degree polynomials, the method benefits from a much higher speed. Finally in our tests on a large corpus of real expressive speech, we show that the proposed method has the potential to be used for voice quality analysis.

The Re-Encoding Transformation in Algebraic List-Decoding of Reed–Solomon Codes

The main computational steps in algebraic soft-decoding, as well as Sudan-type list-decoding, of Reed–Solomon codes are bivariate polynomial interpolation and factorization. We introduce a computational technique, based upon re-encoding and coordinate transformation, that significantly reduces the complexity of the bivariate interpolation procedure. This re-encoding and coordinate transformation converts the original interpolation problem into another reduced interpolation problem, which is orders of magnitude smaller than the original one. A formal proof is presented to show that the two interpolation problems are indeed equivalent. An efficient factorization procedure that applies directly to the reduced interpolation problem is also given.

VLSI Architectures for Soft-Decision Decoding of Reed–Solomon Codes

Soft-decision decoding of Reed-Solomon codes delivers significant coding gains over classical minimum distance decoding. In this paper, we present architectures for polynomial interpolation and factorization, the two main steps of the soft-decoding algorithm. We introduce an algorithmic transformation for reducing the iterations required in generating the interpolation polynomial and present efficient architectures by sharing computations. We also describe algorithmic transformations for further reducing the interpolation and factorization latency. An area efficient, folded-pipelined version of the interpolation architecture is also described. Finally, we present an example of a Reed-Solomon soft decoder utilizing the presented architectures, having a 250 Mbps throughput.

Factorization of polynomials in Z-graded skew polynomial rings

Factorization of polynomials in Z-graded skew polynomial rings

Real solution isolation with multiplicity of zero-dimensional triangular systems

Existing algorithms for isolating real solutions of zero-dimensional polynomial systems do not compute the multiplicities of the solutions. In this paper, we define in a natural way the multiplicity of solutions of zero-dimensional triangular polynomial systems and prove that our definition is equivalent to the classical definition of local (intersection) multiplicity. Then we present an effective and complete algorithm for isolating real solutions with multiplicities of zero-dimensional triangular polynomial systems using our definition. The algorithm is based on interval arithmetic and square-free factorization of polynomials with real algebraic coefficients. The computational results on some examples from the literature are presented.

A new combinatorial interpretation of a $q$-analogue of the Lah numbers

The Lah numbers L(n, k) are the connection constants between the rising factorial and falling factorial polynomial bases and count partitions of n distinct objects into k blocks, where objects within a block are ordered (termed Laguerre configurations). In this paper, we consider the q-Lah numbers defined as the connection constants between the comparable bases of polynomials gotten by replacing each positive integer n with nq = 1+ q + · · ·+ q n−1 and provide a new combinatorial interpretation for them by describing a statistic on Laguerre configurations for which they are the generating function. We describe some other algebraic properties of these numbers and can provide combinatorial explanations in several instances using our interpretation. A further generalization involving a second parameter may also be given.

Testing Fourier Dimensionality and Sparsity

We present a range of new results for testing properties of Boolean functions that are defined in terms of the Fourier spectrum. Broadly speaking, our results show that the property of a Boolean function having a concise Fourier representation is locally testable. We give the first efficient algorithms for testing whether a Boolean function has a sparse Fourier spectrum (small number of nonzero coefficients) and for testing whether the Fourier spectrum of a Boolean function is supported in a low-dimensional subspace of $\mathbb{F}_2^n$. In both cases we also prove lower bounds showing that any testing algorithm—even an adaptive one—must have query complexity within a polynomial factor of our algorithms, which are nonadaptive. Building on these results, we give an “implicit learning” algorithm that lets us test any subproperty of Fourier concision. We also present some applications of these results to exact learning and decoding. Our technical contributions include new structural results about sparse Boolean functions and new analysis of the pairwise independent hashing of Fourier coefficients from [V. Feldman, P. Gopalan, S. Khot, and A. Ponnuswami, Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2006, pp. 563-576].

Fast Polynomial Factorization and Modular Composition

We obtain randomized algorithms for factoring degree $n$ univariate polynomials over $\mathbb{F}_q$ requiring $O(n^{1.5 + o(1)}\,{\rm log}^{1+o(1)} q+ n^{1 + o(1)}\,{\rm log}^{2+o(1)} q)$ bit operations. When ${\rm log}\, q < n$, this is asymptotically faster than the best previous algorithms [J. von zur Gathen and V. Shoup, Comput. Complexity, 2 (1992), pp. 187-224; E. Kaltofen and V. Shoup, Math. Comp., 67 (1998), pp. 1179-1197]; for ${\rm log}\, q \ge n$, it matches the asymptotic running time of the best known algorithms. The improvements come from new algorithms for modular composition of degree $n$ univariate polynomials, which is the asymptotic bottleneck in fast algorithms for factoring polynomials over finite fields. The best previous algorithms for modular composition use $O(n^{(\omega + 1)/2})$ field operations, where $\omega$ is the exponent of matrix multiplication [R. P. Brent and H. T. Kung, J. Assoc. Comput. Mach., 25 (1978), pp. 581-595], with a slight improvement in the exponent achieved by employing fast rectangular matrix multiplication [X. Huang and V. Y. Pan, J. Complexity, 14 (1998), pp. 257-299]. We show that modular composition and multipoint evaluation of multivariate polynomials are essentially equivalent, in the sense that an algorithm for one achieving exponent $\alpha$ implies an algorithm for the other with exponent $\alpha + o(1)$, and vice versa. We then give two new algorithms that solve the problem near-optimally: an algebraic algorithm for fields of characteristic at most $n^{o(1)}$, and a nonalgebraic algorithm that works in arbitrary characteristic. The latter algorithm works by lifting to characteristic 0, applying a small number of rounds of multimodular reduction, and finishing with a small number of multidimensional FFTs. The final evaluations are reconstructed using the Chinese remainder theorem. As a bonus, this algorithm produces a very efficient data structure supporting polynomial evaluation queries, which is of independent interest. Our algorithms use techniques that are commonly employed in practice, in contrast to all previous subquadratic algorithms for these problems, which relied on fast matrix multiplication.

Degree Bounded Network Design with Metric Costs

Given a complete undirected graph, a cost function on edges, and a degree bound $B$, the degree bounded network design problem is to find a minimum cost simple subgraph with maximum degree $B$ satisfying given connectivity requirements. Even for a simple connectivity requirement such as finding a spanning tree, computing a feasible solution for the degree bounded network design problem is already NP-hard, and thus there is no polynomial factor approximation algorithm for this problem. In this paper, we show that when the cost function satisfies the triangle inequality, there are constant factor approximation algorithms for various degree bounded network design problems. In global edge-connectivity, there is a $(2+\frac{1}{k})$-approximation algorithm for the minimum bounded degree $k$-edge-connected subgraph problem. In local edge-connectivity, there is a 4-approximation algorithm for the minimum bounded degree Steiner network problem when $r_{\max}$ is even, and a 5.5-approximation algorithm when $r_{\max}$ is odd, where $r_{\max}$ is the maximum connectivity requirement. In global vertex-connectivity, there is a $(2+\frac{k-1}{n}+\frac{1}{k})$-approximation algorithm for the minimum bounded degree $k$-vertex-connected subgraph problem when $n\geq2k$, where $n$ is the number of vertices. For spanning tree, there is a $(1+\frac{1}{B-1})$-approximation algorithm for the minimum bounded degree spanning tree problem. These approximation algorithms return solutions with the smallest possible maximum degree, and in most cases the cost guarantee is obtained by comparing to the optimal cost when there are no degree constraints. This demonstrates that degree constraints can be incorporated into network design problems with metric costs. Our algorithms can be seen as a generalization of Christofides' algorithm for the metric traveling salesman problem. The main technical tool is a simplicity-preserving edge splitting-off operation, which is used to “short-cut” vertices with high degree while maintaining connectivity requirements and preserving simplicity of the solutions.

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.

A note on Gao’s algorithm for polynomial factorization

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.

Irreducibility criteria of Schur-type and Pólya-type

Let $${f(x)=(x-a_1)\cdots (x-a_m)}$$ , where a 1, . . . , a m are distinct rational integers. In 1908 Schur raised the question whether f(x) ± 1 is irreducible over the rationals. One year later he asked whether $${(f(x))^{2^k}+1}$$ is irreducible for every k ≥ 1. In 1919 Polya proved that if $${P(x)\in\mathbb{Z}[x]}$$ is of degree m and there are m rational integer values a for which 0 < |P(a)| < 2−N N! where $${N=\lceil m/2\rceil}$$ , then P(x) is irreducible. A great number of authors have published results of Schur-type or Polya-type afterwards. Our paper contains various extensions, generalizations and improvements of results from the literature. To indicate some of them, in Theorem 3.1 a Polya-type result is established when the ground ring is the ring of integers of an arbitrary imaginary quadratic number field. In Theorem 4.1 we describe the form of the factors of polynomials of the shape h(x) f(x) + c, where h(x) is a polynomial and c is a constant such that |c| is small with respect to the degree of h(x) f(x). We obtain irreducibility results for polynomials of the form g(f(x)) where g(x) is a monic irreducible polynomial of degree ≤ 3 or of CM-type. Besides elementary arguments we apply methods and results from algebraic number theory, interpolation theory and diophantine approximation.

On the unimodality of independence polynomials of some graphs

In this paper we study unimodality problems for the independence polynomial of a graph, including unimodality, log-concavity and reality of zeros. We establish recurrence relations and give factorizations of independence polynomials for certain classes of graphs. As applications we settle some unimodality conjectures and problems.

General polynomial factorization-based design of sparse periodic linear arrays

We have developed several methods of designing sparse periodic arrays based upon the polynomial factorization method. In these methods, transmit and receive aperture polynomials are selected such that their product results in a polynomial representing the desired combined transmit/receive (T/R) effective aperture function. A desired combined T/R effective aperture is simply an aperture with an appropriate width exhibiting a spectrum that corresponds to the desired two-way radiation pattern. At least one of the two aperture functions that constitute the combined T/R effective aperture function will be a sparse polynomial. A measure of sparsity of the designed array is defined in terms of the element reduction factor. We show that elements of a linear array can be reduced with varying degrees of beam mainlobe width to sidelobe reduction properties.

An Explicit Treatment of Cubic Function Fields with Applications

AbstractWe give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few square-free polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.

Estimating hybrid frequency moments of data streams

We consider the problem of estimating hybrid frequency moments of two dimensional data streams. In this model, data is viewed to be organized in a matrix form (A i,j )1?i,j,?n . The entries A i,j are updated coordinate-wise, in arbitrary order and possibly multiple times. The updates include both increments and decrements to the current value of A i,j . The hybrid frequency moment F p,q (A) is defined as $\sum_{j=1}^{n}(\sum_{i=1}^{n}{A_{i,j}}^{p})^{q}$ and is a generalization of the frequency moment of one-dimensional data streams. We present the first $\tilde{O}(1)$ space algorithm for the problem of estimating F p,q for p?[0,2] and q?[0,1] to within an approximation factor of 1±?. The $\tilde{O}$ notation hides poly-logarithmic factors in the size of the stream m, the matrix size n and polynomial factors of ? ?1. We also present the first $\tilde{O}(n^{1-1/q})$ space algorithm for estimating F p,q for p?[0,2] and q?(1,2].

The minimal polynomial of a sequence obtained from the componentwise linear transformation of a linear recurring sequence

Let S=(s1,s2,…,sm,…) be a linear recurring sequence with terms in GF(qn) and T be a linear transformation of GF(qn) over GF(q). Denote T(S)=(T(s1),T(s2),…,T(sm),…). In this paper, we first present counter examples to show that the main result in [A.M. Youssef, G. Gong, On linear complexity of sequences over GF(2n), Theoret. Comput. Sci., 352 (2006) 288–292] is not correct in general since Lemma 3 in that paper is incorrect. Then we determine the minimal polynomial of T(S) if the canonical factorization of the minimal polynomial of S without multiple roots is known and thus present the solution to the main problem which was considered in the above paper but incorrectly solved. Additionally, as a special case, we determine the minimal polynomial of T(S) if the minimal polynomial of S is primitive. Finally, we give an upper bound on the linear complexity of T(S) when T exhausts all possible linear transformations of GF(qn) over GF(q). This bound is tight in some cases.

A lifting and recombination algorithm for rational factorization of sparse polynomials

We propose a new lifting and recombination scheme for rational bivariate polynomial factorization that takes advantage of the Newton polytope geometry. We obtain a deterministic algorithm that can be seen as a sparse version of an algorithm of Lecerf, with a polynomial complexity in the volume of the Newton polytope. We adopt a geometrical point of view, the main tool being derived from some algebraic osculation criterion in toric varieties.

Towards an ‘average’ version of the Birch and Swinnerton-Dyer conjecture

Text The Birch and Swinnerton-Dyer conjecture states that the rank of the Mordell–Weil group of an elliptic curve E equals the order of vanishing at the central point of the associated L-function L ( s , E ) . Previous investigations have focused on bounding how far we must go above the central point to be assured of finding a zero, bounding the rank of a fixed curve or on bounding the average rank in a family. Mestre (1986) [Mes] showed the first zero occurs by O ( 1 / log log N E ) , where N E is the conductor of E, though we expect the correct scale to study the zeros near the central point is the significantly smaller 1 / log N E . We significantly improve on Mestre's result by averaging over a one-parameter family of elliptic curves E over Q ( T ) . We assume GRH, Tate's conjecture if E is not a rational surface, and either the ABC or the Square-Free Sieve Conjecture if the discriminant has an irreducible polynomial factor of degree at least 4. We find non-trivial upper and lower bounds for the average number of normalized zeros in intervals on the order of 1 / log N E (which is the expected scale). Our results may be interpreted as providing further evidence in support of the Birch and Swinnerton-Dyer conjecture, as well as the Katz–Sarnak density conjecture from random matrix theory (as the number of zeros predicted by random matrix theory lies between our upper and lower bounds). These methods may be applied to additional families of L-functions. Video For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=3EVYPNi_LG0.