Existence Of Polynomials Research Articles

Parameterized Complexity of Independent Set in H-Free Graphs

In this paper, we investigate the complexity of Maximum Independent Set (MIS) in the class of H-free graphs, that is, graphs excluding a fixed graph as an induced subgraph. Given that the problem remains NP-hard for most graphs H, we study its fixed-parameter tractability and make progress towards a dichotomy between FPT and W[1]-hard cases. We first show that MIS remains W[1]-hard in graphs forbidding simultaneously $$K_{1, 4}$$ , any finite set of cycles of length at least 4, and any finite set of trees with at least two branching vertices. In particular, this answers an open question of Dabrowski et al. concerning $$C_4$$ -free graphs. Then we extend the polynomial algorithm of Alekseev when H is a disjoint union of edges to an FPT algorithm when H is a disjoint union of cliques. We also provide a framework for solving several other cases, which is a generalization of the concept of iterative expansion accompanied by the extraction of a particular structure using Ramsey’s theorem. Iterative expansion is a maximization version of the so-called iterative compression. We believe that our framework can be of independent interest for solving other similar graph problems. Finally, we present positive and negative results on the existence of polynomial (Turing) kernels for several graphs H.

On the Darboux integrability of a three—dimensional forced—damped differential system

In 2011 Pehlivan proposed a three—dimensional forced—damped autonomous differential system which can display simultaneously unbounded and chaotic solutions. This untypical phenomenon has been studied recently by several authors. In this paper we study the opposite to its chaotic motion, i.e. its integrability, mainly the existence of polynomial, rational and Darboux first integrals through the analysis of its invariant algebraic surfaces and its exponential factors.

Hypercontractive inequalities via SOS, and the Frankl--Rödl graph

Hypercontractive inequalities via SOS, and the Frankl-Rodl graph, Discrete Analysis 2016:4, 20 pp. One of the major unsolved problems in theoretical computer science is the unique games conjecture of Subhash Khot. Its importance lies in the fact that it would imply that many of our best known approximation algorithms are optimal (assuming that P$\ne$NP). As with the statement that P$\ne$NP, many results have now been proved assuming the unique-games conjecture. However, theoretical computer scientists are much less confident that it is true than they (mostly) are that P$\ne$NP. A possible challenge to the unique games conjecture comes from the so-called sum-of-squares algorithm, which for some problems has turned out to be better than the algorithms based on semidefinite programming that had previously been thought likely to be best possible. Suppose we wish to prove that a system of real polynomial equations $P_1=\dots=P_m=0$ in $k$ variables cannot be simultaneously solved. It can be shown that this is equivalent to the existence of polynomials $Q_1,\dots,Q_m$ and $S$ such that $S$ is a sum of squares of polynomials and $\sum_iP_iQ_i+S=-1$. More interestingly from a computational point of view, there are often situations where the polynomials $Q_i$ and $S$ have low degree, and when that is the case they can be found efficiently. Furthermore, many interesting problems can be reformulated as polynomial equations of this type. In particular, Barak, Brandao, Harrow, Kelner, Steurer and Zhou found an SOS proof of a hypercontractive inequality. Hypercontractive inequalities are bounds for the norms of certain convolution operators from $L_p$ to $L_q$, where the ground sets of these spaces are the discrete cube: the proof obtained by Barak et al was for $p=2$ and $q=4$. They have many applications, in particular to proving _integrality gaps_: that is, optimization problems with significantly different bounds if the answers are required to be integers. If the inequalities can be proved by SOS methods, then that raises the possibility of finding SOS certifications of integrality gaps, which has been achieved for several problems. Although none of these applications is yet powerful enough to refute the unique games conjecture, it is still a promising line of enquiry. In this paper the authors obtain a sum-of-squares proof for the hypercontractivity inequality with $p=2$ and $q=2k$ for any integer $k$. They also find a sum-of-squares proof for a reverse hypercontractive inequality. One consequence of the main results is a degree-4 SOS certification that a certain graph defined by Frankl and Rodl has unbounded chromatic number, despite the fact that standard semidefinite linear programming relaxations of the problem cannot even certify a lower bound better than 3.

General tree-level amplitudes by factorization limits

To find boundary contributions is a rather difficult problem when applying the BCFW recursion relation. In this paper, we propose an approach to bypass this problem by calculating general tree amplitudes that contain no polynomial using factorization limits. More explicitly, we construct an expression iteratively, which produces correct factorization limits for all physical poles, and does not contain other poles, then it should be the correct amplitude. To some extent, this approach can be considered as an alternative way to find boundary contributions. To demonstrate our approach, we present several examples: $\phi^4$ theory, pure gauge theory, Einstein-Maxwell theory, and Yukawa theory. While the amplitude allows the existence of polynomials which satisfy correct mass dimension and helicities, this approach is not applicable to determine the full amplitude.

Using Elimination Theory to Construct Rigid Matrices

The rigidity of a matrix A for target rank r is the minimum number of entries of A that must be changed to ensure that the rank of the altered matrix is at most r. Since its introduction by Valiant (1977), rigidity and similar rank-robustness functions of matrices have found numerous applications in circuit complexity, communication complexity, and learning complexity. Almost all nxn matrices over an infinite field have a rigidity of (n-r)^2. It is a long-standing open question to construct infinite families of explicit matrices even with superlinear rigidity when r = Omega(n). In this paper, we construct an infinite family of complex matrices with the largest possible, i.e., (n-r)^2, rigidity. The entries of an n x n matrix in this family are distinct primitive roots of unity of orders roughly exp(n^2 log n). To the best of our knowledge, this is the first family of concrete (but not entirely explicit) matrices having maximal rigidity and a succinct algebraic description. Our construction is based on elimination theory of polynomial ideals. In particular, we use results on the existence of polynomials in elimination ideals with effective degree upper bounds (effective Nullstellensatz). Using elementary algebraic geometry, we prove that the dimension of the affine variety of matrices of rigidity at most k is exactly n^2-(n-r)^2+k. Finally, we use elimination theory to examine whether the rigidity function is semi-continuous.

Existence of polynomials on subspaces without extension

We prove the existence of a polynomial of degree d d defined on a closed subspace that cannot be extended to the Banach space E E (in particular, the existence of a nonextendible polynomial) in the following cases: (1) d ≥ 2 d\geq 2 and E E does not have type p p for some 1 > p > 2 1>p>2 ; (2) the space ℓ k \ell _k , k ∈ N k\in \mathbb {N} , 2 > k ≤ d 2>k\leq d , is finitely representable in E E . In each of these cases we prove, equivalently, the existence of a closed subspace F ⊂ E F\subset E such that the subspace ⊗ ^ s , π d F \hat {\otimes }^{d}_{s,\pi }{F} is not closed in ⊗ ^ s , π d E \hat {\otimes }^{d}_{s,\pi }{E} .

On the Existence of Polynomials with Chaotic Behaviour

We establish a general result on the existence of hypercyclic (resp., transitive, weakly mixing, mixing, frequently hypercyclic) polynomials on locally convex spaces. As a consequence we prove that every (real or complex) infinite-dimensional separable Frèchet space admits mixing (hence hypercyclic) polynomials of arbitrary positive degree. Moreover, every complex infinite-dimensional separable Banach space with an unconditional Schauder decomposition and every complex Frèchet space with an unconditional basis support chaotic and frequently hypercyclic polynomials of arbitrary positive degree. We also study distributional chaos for polynomials and show that every infinite-dimensional separable Banach space supports polynomials of arbitrary positive degree that have a dense distributionally scrambled linear manifold.

On rational integrability of Euler equations on Lie algebra so(4, ℂ)

We consider the Euler equations on the Lie algebra so(4, ℂ) with a diagonal quadratic Hamiltonian. It is known that this system always admits three functionally independent polynomial first integrals. We prove that if the system has a rational first integral functionally independent of the known three ones so called fourth integral, then it has a polynomial first integral that is also functionally independent of them. This is a consequence of more general fact that for these systems the existence of Darboux polynomial with no vanishing cofactor implies the existence of polynomial fourth integral.

Singular polynomials and modules for the symmetric groups

For certain negative rational numbers k0, called singular values, and associated with the symmetric group S_N on N objects, there exist homogeneous polynomials annihilated by each Dunkl operator when the parameter k = k0. It was shown by de Jeu, Opdam and the author (TAMS 346(1994),237-256) that the singular values are exactly the values m/n with 2<=n<=N, m = 1,2... and m/n is not an integer. For each pair (m,n) satisfying these conditions there is a unique irreducible S_N-module of singular polynomials for the singular value -m/n. The existence of these polynomials was established by the author (IMRN 2004,#67,3607-3635). The uniqueness is proven in the present paper. By using Murphy's (J. Alg. 69(1981), 287-297) results on the eigenvalues of the Murphy elements, the problem of existence of singular polynomials is first restricted to the isotype of a partition of N (corresponding to an irreducible representation of S_N) such that (n/gcd(m,n)) divides t+1 for each part t of the partition except the last one. Then by arguments involving nonsymmetric Jack polynomials it is shown that the assumption that the second part of the partition is greater than or equal to n/gcd(m,n) leads to a contradiction.This shows that the singular polynomials are exactly those already determined.

Mathematical Methods of Regular Dynamics (Kowalevski Workshop)

Mathematical Methods of Regular Dynamics (Kowalevski Workshop)

On the sharpness of an asymptotic error estimate for conjugate gradients

Recently, the authors obtained an upper bound on the error for the conjugate gradient method, which is valid in an asymptotic setting as the size of the linear systems tends to infinity. The estimate depends on the asymptotic distribution of eigenvalues, and the ratio between the size and the number of iterations. Such error bounds are related to the existence of polynomials with value 1 at 0 whose supnorm on the spectrum is as small as possible. A possible strategy for constructing such a polynomial p is to select a set S, to specify that every eigenvalue outside S is a zero of p, and then to minimize the supnorm of p on S. Here we show that this strategy can never give a better asymptotic upper bound than the one we obtained before. We also discuss the case where equality is met.

Describing quadratic Cremer point polynomials by parabolic perturbations

We describe two infinite-order parabolic perturbation procedures yielding quadratic polynomials having a Cremer fixed point. The main idea is to obtain the polynomial as the limit of repeated parabolic perturbations. The basic tool at each step is to control the behaviour of certain external rays.Polynomials of the Cremer type correspond to parameters at the boundary of a hyperbolic component of the Mandelbrot set. In this paper we concentrate on the main cardioid component. We investigate the differences between two-sided (i.e. alternating) and one-sided parabolic perturbations.In the two-sided case, we prove the existence of polynomials having an explicitly given external ray accumulating both at the Cremer point and at its non-periodic preimage. We think of the Julia set as containing a ‘topologist's double comb’.In the one-sided case we prove a weaker result: the existence of polynomials having an explicitly given external ray accumulating at the Cremer point, but having in the impression of the ray both the Cremer point and its other preimage. We think of the Julia set as containing a ‘topologist's single comb’.By tuning, similar results hold on the boundary of any hyperbolic component of the Mandelbrot set.

The Tunnel Diode as a Linear Network Element

Theorems are proved which completely characterize in an explicit manner the class of immittance matrices realizable with lossless reciprocal elements and a tunnel diode represented by the three-parameter “LC, — R” model. Techniques are presented for the synthesis of any immittance matrix within this class. Considered first, from a scattering matrix viewpoint, are the so-called degenerate cases in which the immittance matrices of the lossless network do not exist. Throughout the remainder of the discussion it is assumed that the lossless network possesses an immittance matrix. Necessary and sufficient conditions, involving in a complicated manner the existence of a certain strict Hurwitz polynomial, are derived for realization with a wide class of terminations. A study of the existence of this polynomial for the particular terminations of interest leads to explicit realizability conditions.