Certain Families Of Polynomials Research Articles

On some $$\Sigma ^{B}_{0}$$-formulae generalizing counting principles over $$V^{0}$$

AbstractWe formalize various counting principles and compare their strengths over $$V^{0}$$ V 0 . In particular, we conjecture the following mutual independence between: a uniform version of modular counting principles and the pigeonhole principle for injections, a version of the oddtown theorem and modular counting principles of modulus p, where p is any natural number which is not a power of 2, and a version of Fisher’s inequality and modular counting principles. Then, we give sufficient conditions to prove them. We give a variation of the notion of PHP-tree and k-evaluation to show that any Frege proof of the pigeonhole principle for injections admitting the uniform counting principle as an axiom scheme cannot have o(n)-evaluations. As for the remaining two, we utilize well-known notions of p-tree and k-evaluation and reduce the problems to the existence of certain families of polynomials witnessing violations of the corresponding combinatorial principles with low-degree Nullstellensatz proofs from the violation of the modular counting principle in concern.

Exact Solution of Interacting Particle Systems Related to Random Matrices

We consider one-dimensional diffusions, with polynomial drift and diffusion coefficients, so that in particular the motion can be space-inhomogeneous, interacting via one-sided reflections. The prototypical example is the well-known model of Brownian motions with one-sided collisions, also known as Brownian TASEP, which is equivalent to Brownian last passage percolation. We obtain a formula for the finite dimensional distributions of these particle systems, starting from arbitrary initial condition, in terms of a Fredholm determinant of an explicit kernel. As far as we can tell, in the space-inhomogeneous setting and for general initial condition this is the first time such a result has been proven. We moreover consider the model of non-colliding diffusions, again with polynomial drift and diffusion coefficients, which includes the ones associated to all the classical ensembles of random matrices. We prove that starting from arbitrary initial condition the induced point process has determinantal correlation functions in space and time with an explicit correlation kernel. A key ingredient in our general method of exact solution for both models is the application of the backward in time diffusion flow on certain families of polynomials constructed from the initial condition.

Finite free cumulants: Multiplicative convolutions, genus expansion and infinitesimal distributions

Given two polynomials p ( x ) , q ( x ) p(x), q(x) of degree d d , we give a combinatorial formula for the finite free cumulants of p ( x ) ⊠ d q ( x ) p(x)\boxtimes _d q(x) . We show that this formula admits a topological expansion in terms of non-crossing multi-annular permutations on surfaces of different genera. This topological expansion, on the one hand, deepens the connection between the theories of finite free probability and free probability, and in particular proves that ⊠ d \boxtimes _d converges to ⊠ \boxtimes as d d goes to infinity. On the other hand, borrowing tools from the theory of second order freeness, we use our expansion to study the infinitesimal distribution of certain families of polynomials which include Hermite and Laguerre, and draw some connections with the theory of infinitesimal distributions for real random matrices. Finally, building on our results we give a new short and conceptual proof of a recent result (see J. Hoskins and Z. Kabluchko [Exp. Math. (2021), pp. 1–27]; S. Steinerberger [Exp. Math. (2021), pp. 1–6]) that connects root distributions of polynomial derivatives with free fractional convolution powers.

Algebraic Number Starscapes

We study the geometry of algebraic numbers in the complex plane, and their Diophantine approximation, aided by extensive computer visualization. Motivated by the resulting images, which we have called algebraic starscapes, we describe the geometry of the map from the coefficient space of polynomials to the root space, focusing on the quadratic and cubic cases. The geometry describes and explains the notable features of the illustrations, and motivates a geometric-minded recasting of fundamental results in the Diophantine approximation of the complex plane. Meanwhile, the images provide a case-study in the symbiosis of illustration and research, and an entry-point to geometry and number theory for a wider audience. In particular, the paper is written to provide an accessible introduction to the study of homogeneous geometry and Diophantine approximation. We investigate the homogeneous geometry of root and coefficient spaces under the natural action. Hyperbolic geometry and the discriminant play an important role in low degree. In particular, we rediscover the quadratic and cubic root formulas as isometries of and its unit tangent bundle, respectively. Utilizing this geometry, we determine when the map sending certain families of polynomials to their complex roots (our starscape images) are embeddings. We reconsider the fundamental questions of the Diophantine approximation of complex numbers by algebraic numbers of bounded degree, from the geometric perspective developed. In the quadratic case (approximation by quadratic irrationals), we consider approximation in terms of hyperbolic distance between roots in the complex plane and the discriminant as a measure of arithmetic height on a polynomial. In particular, we determine the supremum on the exponent k for which an algebraic target α has infinitely many approximations β whose hyperbolic distance from α does not exceed . It turns out to fall into two cases, depending on whether α lies on the image of a plane of rational slope in coefficient space (a rational geodesic). The result comes as an application of Schmidt’s subspace theorem. Our results recover the quadratic case of results of Bugeaud and Evertse, and give some geometric explanation for the dichotomy they discovered. Our statements go a little further in distinguishing approximability in terms of whether the target or approximations lie on rational geodesics. The paper comes with accompanying software, and finishes with a wide variety of open problems.

Summand minimality and asymptotic convergence of generalized Zeckendorf decompositions

Given a recurrence sequence $H$, with $H_n = c_1 H_{n-1} + \dots + c_t H_{n-t}$ where $c_i \in \mathbb{N}_0$ for all $i$ and $c_1, c_t \geq 1$, the generalized Zeckendorf decomposition (gzd) of $m \in \mathbb{N}_0$ is the unique representation of $m$ using $H$ composed of blocks lexicographically less than $\sigma = (c_1, \dots, c_t)$. We prove that the gzd of $m$ uses the fewest number of summands among all representations of $m$ using $H$, for all $m$, if and only if $\sigma$ is weakly decreasing. We develop an algorithm for moving from any representation of $m$ to the gzd, the analysis of which proves that $\sigma$ weakly decreasing implies summand minimality. We prove that the gzds of numbers of the form $v_0 H_n + \dots + v_\ell H_{n-\ell}$ converge in a suitable sense as $n \to \infty$, furthermore we classify three distinct behaviors for this convergence. We use this result, together with the irreducibility of certain families of polynomials, to exhibit a representation with fewer summands than the gzd if $\sigma$ is not weakly decreasing.

A note on families of monogenic number fields

We give a sufficient criterion for specializations of certain families of polynomials to yield monogenic number fields. This generalizes constructions in several earlier papers. As applications we give new infinite families of monogenic number fields for several prescribed Galois groups.

Certain families of polynomials arising in the study of hyperelliptic Lie algebras

The commutative ring $$R(P(t))=\mathbb C[t^{\pm 1},u \,|\, u^2=P(t)]$$ , where $$P(t)=\sum _{i=0}^na_it^i=\prod _{k=1}^n(t-\alpha _i)$$ with $$\alpha _i\in \mathbb C$$ pairwise distinct, is the coordinate ring of a hyperelliptic curve when $$n>4$$ . The Lie algebra $$\mathcal {R}(P(t))={\text {Der}}(R(P(t)))$$ of derivations is called the hyperelliptic Lie algebra associated to P(t) and is a particular type of multipoint Krichever–Novikov algebra. In this paper, we describe the universal central extension of $${\text {Der}}(R(P(t)))$$ in terms of certain families of polynomials which in a particular case are associated Legendre polynomials. Moreover we describe certain families of polynomials that arise in the study of the group of units for the ring R(P(t)), where $$P(t)=t^4-2bt^2+1$$ . In this study, pairs of Chebyshev polynomials $$(U_n,T_n)$$ arise as particular cases of a pairs $$(r_n,s_n)$$ with $$r_n+s_n\sqrt{P(t)}$$ a unit in R(P(t)). We explicitly describe these polynomial pairs as coefficients of certain generating functions and show that certain of these polynomials satisfy particular second-order linear differential equations.

Computing Galois groups of certain families of polynomials

Computing Galois groups of certain families of polynomials

Distinct distances on regular varieties over finite fields

In this paper we study some generalized versions of a recent result due to Covert, Koh, and Pi (2015). More precisely, we prove that if a subset E in a regular variety satisfies |E|≫qd−12+1k−1, thenΔk,F(E):={F(x1+⋯+xk):xi∈E,1≤i≤k}⊇Fq∖{0}, for some certain families of polynomials F(x)∈Fq[x1,…,xd].

The Bohnenblust-Hille inequality combined with an inequality of Helson

We give a variant of the Bohenblust-Hille inequality which, for certain families of polynomials, leads to constants with polynomial growth in the degree.

Convergence of polynomial ergodic averages

We prove theL2 convergence for an ergodic average of a product of functions evaluated along polynomial times in a totally ergodic system. For each set of polynomials, we show that there is a particular factor, which is an inverse limit of nilsystems, that controls the limit behavior of the average. For a general system, we prove the convergence for certain families of polynomials.

Constructions for Permutation Codes in Powerline Communications

A permutation array (or code) of length n and distance d is a set Γ of permutations from some fixed set of n symbols such that the Hamming distance between each distinct x, y ∈ Γ is at least d. One motivation for coding with permutations is powerline communication. After summarizing known results, it is shown here that certain families of polynomials over finite fields give rise to permutation arrays. Additionally, several new computational constructions are given, often making use of automorphism groups. Finally, a recursive construction for permutation arrays is presented, using and motivating the more general notion of codes with constant weight composition.

Alexander polynomials of non-locally-flat knots

We generalize the classical study of Alexander polynomials of smooth or PL locally-flat knots to PL knots that are not necessarily locally-flat. We introduce three families of generalized Alexander polynomials and study their properties. For knots with point singularities, we obtain a classification of these polynomials that is complete except for one special low-dimensional case. This classification extends existing classifications for PL locally-flat knots. For knots with higher-dimensional singularities, we further extend the necessary conditions on the invariants. We also construct several varieties of singular knots to demonstrate realizability of certain families of polynomials as generalized Alexander polynomials. These constructions, of independent interest, generalize known knot constructions such as frame spinning and twist spinning.

Second order difference equations for certain families of ‘discrete’ polynomials

In this paper we will consider two algorithms which allow us to obtain second order linear difference equations for certain families of polynomials. The corresponding algorithms can be implemented in any computer algebra system in order to obtain explicit expressions of the coefficients of the difference equations.

Societies and Academies

PARIS Academy of Sciences, Aug. 17 (vol. 195, pp. 429–448).—Charles Achard and Ho-dac-an: Some observations on the flocculation of suspensions of myxoprotein by electrolytes. The existence of two zones in the flocculation curve of myxoprotein by the two electrolytes studied, A1C13 and ThCl4, leads provisionally to two hypotheses: myxoprotein contains two different substances or there is a change of sign in the protein.—E. Bataillon and P. Tcherniakofsky: The sterility of the male hybrids resulting from crossing Molge marmorata and Molge cristata.—R. Risser: The proper dispersion with n errors in the case where each of the component errors is ruled by a simple law. An attempt at an analytical representation.—Jacques Devisme: Certain families of polynomials.—D. Iwanenko: The constitution of the atomic nuclei.—A. Portevin and P. Bastien: Contribution to the study of the ternary system magnesium-aluminium-copper.—Li Shi Lin: Study of some ottrelite schists from China.—Henry Germain: Some fresh-water diatoms living in mucous tubes.