Showing papers on "Square-free polynomial published in 2017"

Translation principle for Dirac index

Abstract: Let $G$ be a finite cover of a closed connected transpose-stable subgroup of ${\rm GL}(n,\Bbb{R})$ with complexified Lie algebra ${\frak g}$. Let $K$ be a maximal compact subgroup of $G$, and assume that $G$ and $K$ have equal rank. We prove a translation principle for the Dirac index of virtual $({\frak g},K)$-modules. As a byproduct, to each coherent family of such modules, we attach a polynomial on the dual of the compact Cartan subalgebra of ${\frak g}$. This ``index polynomial'' generates an irreducible representation of the Weyl group contained in the coherent continuation representation. We show that the index polynomial is the exact analogue on the compact Cartan subgroup of King's character polynomial. The character polynomial was defined by King on the maximally split Cartan subgroup, and it was shown to be equal to the Goldie rank polynomial up to a scalar multiple. In the case of representations of Gelfand-Kirillov dimension at most half the dimension of $G/K$, we also conjecture an explicit relationship between our index polynomial and the multiplicities of the irreducible components occurring in the associated cycle of the corresponding coherent family.

Learning linear structural equation models in polynomial time and sample complexity

Abstract: The problem of learning structural equation models (SEMs) from data is a fundamental problem in causal inference. We develop a new algorithm --- which is computationally and statistically efficient and works in the high-dimensional regime --- for learning linear SEMs from purely observational data with arbitrary noise distribution. We consider three aspects of the problem: identifiability, computational efficiency, and statistical efficiency. We show that when data is generated from a linear SEM over $p$ nodes and maximum degree $d$, our algorithm recovers the directed acyclic graph (DAG) structure of the SEM under an identifiability condition that is more general than those considered in the literature, and without faithfulness assumptions. In the population setting, our algorithm recovers the DAG structure in $\mathcal{O}(p(d^2 + \log p))$ operations. In the finite sample setting, if the estimated precision matrix is sparse, our algorithm has a smoothed complexity of $\widetilde{\mathcal{O}}(p^3 + pd^7)$, while if the estimated precision matrix is dense, our algorithm has a smoothed complexity of $\widetilde{\mathcal{O}}(p^5)$. For sub-Gaussian noise, we show that our algorithm has a sample complexity of $\mathcal{O}(\frac{d^8}{\varepsilon^2} \log (\frac{p}{\sqrt{\delta}}))$ to achieve $\varepsilon$ element-wise additive error with respect to the true autoregression matrix with probability at most $1 - \delta$, while for noise with bounded $(4m)$-th moment, with $m$ being a positive integer, our algorithm has a sample complexity of $\mathcal{O}(\frac{d^8}{\varepsilon^2} (\frac{p^2}{\delta})^{1/m})$.

Sparse Rational Univariate Representation

Abstract: We present explicit worst case degree and height bounds for the rational univariate representation of the isolated roots of polynomial systems based on mixed volume. We base our estimations on height bounds of resultants and we consider the case of 0-dimensional, positive dimensional, and parametric polynomial systems.

Factorization of Z-homogeneous polynomials in the First (q)-Weyl Algebra

Abstract: Factorization of elements of noncommutative rings is an important problem both in theory and applications. For the class of domains admitting nontrivial grading, we have recently proposed an approach, which utilizes the grading in order to factor general elements. This is heavily based on the factorization of graded elements. In this paper, we present algorithms to factorize weighted homogeneous (graded) elements in the polynomial first q-Weyl and Weyl algebras, which are both viewed as \({ \mathbb {Z}}\)-graded rings. We show that graded polynomials have finite number of factorizations. Moreover, the factorization of such can be almost completely reduced to commutative univariate factorization over the same base field with some additional uncomplicated combinatorial steps. This allows to deduce the complexity of our algorithms in detail, which we prove to be polynomial-time. Furthermore, we show, that for a graded polynomial p, irreducibility of p in the polynomial first Weyl algebra implies its irreducibility in the localized (rational) Weyl algebra, which is not true for general polynomials. We report on our implementation in the computer algebra system Singular. For graded polynomials, it outperforms currently available implementations for factoring in the first Weyl algebra—in speed as well as in elegancy of the results.

Solving Multivariate Polynomial Systems and an Invariant from Commutative Algebra

Abstract: The complexity of computing the solutions of a system of multivariate polynomial equations by means of Groebner bases computations is upper bounded by a function of the solving degree. In this paper, we discuss how to rigorously estimate the solving degree of a system, focusing on systems arising within public-key cryptography. In particular, we show that it is upper bounded by, and often equal to, the Castelnuovo-Mumford regularity of the ideal generated by the homogenization of the equations of the system, or by the equations themselves in case they are homogeneous. We discuss the underlying commutative algebra and clarify under which assumptions the commonly used results hold. In particular, we discuss the assumption of being in generic coordinates (often required for bounds obtained following this type of approach) and prove that systems that contain the field equations or their fake Weil descent are in generic coordinates. We also compare the notion of solving degree with that of degree of regularity, which is commonly used in the literature. We complement the paper with some examples of bounds obtained following the strategy that we describe.

Computation of Induced Orthogonal Polynomial Distributions

Abstract: We provide a robust and general algorithm for computing distribution functions associated to induced orthogonal polynomial measures. We leverage several tools for orthogonal polynomials to provide a spectrally-accurate method for a broad class of measures, which is stable for polynomial degrees up to at least degree 1000. Paired with other standard tools such as a numerical root-finding algorithm and inverse transform sampling, this provides a methodology for generating random samples from an induced orthogonal polynomial measure. Generating samples from this measure is one ingredient in optimal numerical methods for certain types of multivariate polynomial approximation. For example, sampling from induced distributions for weighted discrete least-squares approximation has recently been shown to yield convergence guarantees with a minimal number of samples. We also provide publicly-available code that implements the algorithms in this paper for sampling from induced distributions.

A simpler and faster strongly polynomial algorithm for generalized flow maximization

Abstract: We present a new strongly polynomial algorithm for generalized flow maximization. The first strongly polynomial algorithm for this problem was given very recently by Vegh; our new algorithm is much simpler, and much faster. The complexity bound O((m+nlogn)mnlog(n2/m)) improves on the previous estimate obtained by Vegh by almost a factor O(n2). Even for small numerical parameter values, our algorithm is essentially as fast as the best weakly polynomial algorithms. The key new technical idea is relaxing primal feasibility conditions. This allows us to work almost exclusively with integral flows, in contrast to all previous algorithms.

Factorization of Polynomials over Z/(pn)

Abstract: In this paper, we deal with the problem of finding a factorization of a monic primary polynomial f ∈ Z/(pn)[x] into irreducible factors. This task has been completely solved when pn does not divide the discriminant of f, while there is not an efficient method of determining a factorization when this happens and finding an explicit factorization can be hard for polynomials of high degree. We discuss some techniques to speed up the computation, focusing on the case n=3.

Continued fractions in function fields: polynomial analogues of McMullen's and Zaremba's conjectures

Abstract: We examine the polynomial analogues of McMullen's and Zaremba's conjectures on continued fractions with bounded partial quotients. It has already been proved by Blackburn that if the base field is infinite, then the polynomial analogue of Zaremba's conjecture holds; we will prove this again with a different method and examine some known results for finite base fields. Translating to the polynomial setting a result of Mercat, we will prove that the polynomial analogue of McMullen's conjecture holds over infinite algebraic extensions of finite fields and that, over finite fields, it would be a consequence of the polynomial analogue of Zaremba's conjecture. We will then prove that the polynomial analogue of McMullen's conjecture holds over uncountable base fields, over $\overline{\mathbb Q}$ (thanks to the theory of reduction of a formal Laurent series modulo a prime) and over number fields. For this purpose, we will examine the connection between the continued fractions of polynomial multiples of $\sqrt D$ and pullbacks of generalized Jacobians of the hyperelliptic curve $U^2=D(T)$.

Polynomial Proof Systems, Effective Derivations, and their Applications in the Sum-of-Squares Hierarchy

Abstract: Author(s): Weitz, Benjamin | Advisor(s): Raghavendra, Prasad | Abstract: Semidenite programming (SDP) relaxations have been a popular choice for approximationalgorithm design ever since Goemans and Williamson used one to improve the best approximationof Max-Cut in 1992. In the effort to construct stronger and stronger SDP relaxations,the Sum-of-Squares (SOS) or Lasserre hierarchy has emerged as the most promisingset of relaxations. However, since the SOS hierarchy is relatively new, we still do not knowthe answer to even very basic questions about its power. For example, we do not even knowwhen the SOS SDP is guaranteed to run correctly in polynomial time!In this dissertation, we study the SOS hierarchy and make positive progress inunderstanding the above question, among others. First, we give a sufficient, simple criteriawhich implies that an SOS SDP will run in polynomial time, as well as confirm that ourcriteria holds for a number of common applications of the SOS SDP. We also present anexample of a Boolean polynomial system which has SOS certificates that require exponential time to find, even though the certificates are degree two. This answers a conjecture of [54].Second, we study the power of the SOS hierarchy relative to other symmetric SDP relaxationsof comparable size. We show that in some situations, the SOS hierarchy achievesthe best possible approximation among every symmetric SDP relaxation. In particular, weshow that the SOS SDP is optimal for the Matching problem. Together with an SOS lowerbound due to Grigoriev [32], this implies that the Matching problem has no subexponentialsize symmetric SDP relaxation. This can be viewed as an SDP analogy of Yannakakis'original symmetric LP lower bound [72].As a key technical tool, our results make use of low-degree certificates of ideal membershipfor the polynomial ideal formed by polynomial constraints. Thus an important step in ourproofs is constructing certificates for arbitrary polynomials in the corresponding constraintideals. We develop a meta-strategy for exploiting symmetries of the underlying combinatorialproblem. We apply our strategy to get low-degree certificates for Matching, Balanced CSP, TSP, and others.

Symbolic Computations of First Integrals for Polynomial Vector Fields.

Abstract: In this article we show how to generalize to the Darbouxian, Liouvillian and Riccati case the extactic curve introduced by J. Pereira. With this approach, we get new algorithms for computing, if it exists, a rational, Darbouxian, Liouvillian or Riccati first integral with bounded degree of a polynomial planar vector field. We give probabilistic and deterministic algorithms. The arithmetic complexity of our probabilistic algorithm is in $\tilde{\mathcal{O}}(N^{\omega+1})$, where $N$ is the bound on the degree of a representation of the first integral and $\omega \in [2;3]$ is the exponent of linear algebra. This result improves previous algorithms. Our algorithms have been implemented in Maple and are available on authors' websites. In the last section, we give some examples showing the efficiency of these algorithms.

A structure theorem for poorly anticoncentrated polynomials of Gaussians and applications to the study of polynomial threshold functions

Abstract: We prove a structural result for degree-$d$ polynomials. In particular, we show that any degree-$d$ polynomial, $p$ can be approximated by another polynomial, $p_{0}$, which can be decomposed as some function of polynomials $q_{1},\ldots,q_{m}$ with $q_{i}$ normalized and $m=O_{d}(1)$, so that if $X$ is a Gaussian random variable, the probability distribution on $(q_{1}(X),\ldots,q_{m}(X))$ does not have too much mass in any small box. Using this result, we prove improved versions of a number of results about polynomial threshold functions, including producing better pseudorandom generators, obtaining a better invariance principle, and proving improved bounds on noise sensitivity.

Law of inertia for the factorization of cubic polynomials - the imaginary case

Abstract: Let D be a square-free positive integer not divisible by 3 such that the class number h(-3D) of Q((-3D)^(1/2)) is also not divisible by 3. We prove that all cubic polynomials f (x) = x^3 + ax^2 + bx + c in Z[x] with a discriminant D have the same type of factorization over any Galois field F_p, where p is a prime bigger than 3. Moreover, we show that any polynomial f(x) with such a discriminant D has a rational integer root. A complete discussion of the case D = 0 is also included.

Krull dimension and unique factorization in Hurwitz polynomial rings

Abstract: Let $R$ be a commutative ring with identity, and let $R[x]$ be the collection of polynomials with coefficients in~$R$. We observe that there are many multiplications in $R[x]$ such that, together with the usual addition, $R[x]$ becomes a ring that contains $R$ as a subring. These multiplications belong to a class of functions $\lambda $ from $\mathbb {N}_0$ to $\mathbb {N}$. The trivial case when $\lambda (i) = 1$ for all $i$ gives the usual polynomial ring. Among nontrivial cases, there is an important one, namely, the case when $\lambda (i) = i!$ for all $i$. For this case, it gives the well-known Hurwitz polynomial ring $R_H[x]$. In this paper, we study Krull dimension and unique factorization in $R_H[x]$. We show in general that $\dim R \leq \dim R_H[x] \leq 2\dim R +1$. When the ring $R$ is Noetherian we prove that $\dim R \leq \dim R_H[x] \leq \dim R+1$. A condition for the ring $R$ is also given in order to determine whether $\dim R_H[x] = \dim R$ or $\dim R_H[x] = \dim R +1$ in this case. We show that $R_H[x]$ is a unique factorization domain, respectively, a Krull domain, if and only if $R$ is a unique factorization domain, respectively, a Krull domain, containing all of the rational numbers.

Computing Simple Multiple Zeros of Polynomial Systems

Abstract: Given a polynomial system f associated with a simple multiple zero x of multiplicity {\mu}, we give a computable lower bound on the minimal distance between the simple multiple zero x and other zeros of f. If x is only given with limited accuracy, we propose a numerical criterion that f is certified to have {\mu} zeros (counting multiplicities) in a small ball around x. Furthermore, for simple double zeros and simple triple zeros whose Jacobian is of normalized form, we define modified Newton iterations and prove the quantified quadratic convergence when the starting point is close to the exact simple multiple zero. For simple multiple zeros of arbitrary multiplicity whose Jacobian matrix may not have a normalized form, we perform unitary transformations and modified Newton iterations, and prove its non-quantified quadratic convergence and its quantified convergence for simple triple zeros.

Reachability Problem for Polynomial Iteration Is PSPACE-complete.

Abstract: In the reachability problem for polynomial iteration, we are given a set of polynomials over integers and we are asked whether a particular integer can be reached by a non-deterministic application of polynomials. This model can be seen as a generalisation of vector addition systems. Our main result is that the problem is PSPACE-complete for single variable polynomials. On the other hand, the problem is undecidable for multidimensional polynomials, already starting with three dimensions.

Direct sum decomposability of polynomials and factorization of associated forms

Abstract: We prove two criteria for direct sum decomposability of homogeneous polynomials. For a homogeneous polynomial with a non-zero discriminant, we interpret direct sum decomposability of the polynomial in terms of factorization properties of the Macaulay inverse system of its Milnor algebra. This leads to an if-and-only-if criterion for direct sum decomposability of such a polynomial, and to an algorithm for computing direct sum decompositions over any field, either of characteristic $0$ or of sufficiently large positive characteristic, for which polynomial factorization algorithms exist. For homogeneous forms over algebraically closed fields, we interpret direct sums and their limits as forms that cannot be reconstructed from their Jacobian ideal. We also give simple necessary criteria for direct sum decomposability of arbitrary homogeneous polynomials over arbitrary fields and apply them to prove that many interesting classes of homogeneous polynomials are not direct sums.

Algebras and valuations related to the Tutte polynomial

Abstract: This is a chapter destined for the book "Handbook of the Tutte Polynomial". The chapter is a composite. The first part is a brief introduction to Orlik-Solomon algebras. The second part sketches the theory of evaluative functions on matroid base polytopes and in particular, the G-invariant (as the subject is known late 2015). A third very short section is on Hopf-algebra or coalgebra structures in Tutte polynomial theory.

Multilevel weighted least squares polynomial approximation

Abstract: Weighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that, using an optimal distribution of sample locations, the number of samples required to achieve quasi-optimal approximation in a given polynomial subspace scales, up to a logarithmic factor, linearly in the dimension of this space. However, in many applications, the computation of samples includes a numerical discretization error. Thus, obtaining polynomial approximations with a single level method can become prohibitively expensive, as it requires a sufficiently large number of samples, each computed with a sufficiently small discretization error. As a solution to this problem, we propose a multilevel method that utilizes samples computed with different accuracies and is able to match the accuracy of single-level approximations with reduced computational cost. We derive complexity bounds under certain assumptions about polynomial approximability and sample work. Furthermore, we propose an adaptive algorithm for situations where such assumptions cannot be verified a priori. Finally, we provide an efficient algorithm for the sampling from optimal distributions and an analysis of computationally favorable alternative distributions. Numerical experiments underscore the practical applicability of our method.

About some UP-based polynomial fragments of SAT

Abstract: This paper explores several polynomial fragments of SAT that are based on the unit propagation (UP) mechanism. As a first case study, one Tovey's polynomial fragment of SAT is extended through the use of UP. Then, we answer an open question about connections between the so-called UP-Horn class (and other UP-based polynomial variants) and Dalal's polynomial Quad class. Finally, we introduce an extended UP-based pre-processing procedure that allows us to prove that some series of benchmarks from the SAT competitions are polynomial ones. Moreover, our experimentations show that this pre-processing can speed-up the satisfiability check of other instances.

A Stabilized Normal Form Algorithm for Generic Systems of Polynomial Equations

Abstract: We propose a numerical linear algebra based method to find the multiplication operators of the quotient ring $\mathbb{C}[x]/I$ associated to a zero-dimensional ideal $I$ generated by $n$ $\mathbb{C}$-polynomials in $n$ variables. We assume that the polynomials are generic in the sense that the number of solutions in $\mathbb{C}^n$ equals the Bezout number. The main contribution of this paper is an automated choice of basis for $\mathbb{C}[x]/I$, which is crucial for the feasibility of normal form methods in finite precision arithmetic. This choice is based on numerical linear algebra techniques and governed by the numerical properties of the given generators of $I$.

Two-sided space-time $L^1$ approximation and optimal control of polynomial systems

Abstract: We study a two-sided space-time $L^1$ optimization problem and show how to reformulate the problem within the framework of optimal control theory for polynomial systems. This yields insight on the structure of the optimal solution. We prove existence and uniqueness of the optimal solution, and we characterize it by means of the Pontryagin maximum principle. The cost function and the control converge when the polynomial degree tends to $+∞$. We illustrate the theory with numerical simulations, which show that our optimal control interpretation leads to efficient algorithms.

Kato-Milne Cohomology and Polynomial Forms

Abstract: Given a prime number $p$, a field $F$ with $\operatorname{char}(F)=p$ and a positive integer $n$, we study the class-preserving modifications of Kato-Milne classes of decomposable differential forms These modifications demonstrate a natural connection between differential forms and $p$-regular forms A $p$-regular form is defined to be a homogeneous polynomial form of degree $p$ for which there is no nonzero point where all the order $p-1$ partial derivatives vanish simultaneously We define a $\widetilde C_{p,m}$ field to be a field over which every $p$-regular form of dimension greater than $p^m$ is isotropic The main results are that for a $\widetilde C_{p,m}$ field $F$, the symbol length of $H_p^2(F)$ is bounded from above by $p^{m-1}-1$ and for any $n \geq \lceil (m-1) \log_2(p) \rceil+1$, $H_p^{n+1}(F)=0$

Best polynomial approximation on the triangle

Abstract: Let $E_n(f)_{\alpha,\beta,\gamma}$ denote the error of best approximation by polynomials of degree at most $n$ in the space $L^2(\varpi_{\alpha,\beta,\gamma})$ on the triangle $\{(x,y): x, y \ge 0, x+y \le 1\}$, where $\varpi_{\alpha,\beta,\gamma}(x,y) := x^\alpha y ^\beta (1-x-y)^\gamma$ for $\alpha,\beta,\gamma > -1$. Our main result gives a sharp estimate of $E_n(f)_{\alpha,\beta,\gamma}$ in terms of the error of best approximation for higher order derivatives of $f$ in appropriate Sobolev spaces. The result also leads to a characterization of $E_n(f)_{\alpha,\beta,\gamma}$ by a weighted $K$-functional.