Product Of Linear Factors Research Articles

Roots of Descent Polynomials and an Algebraic Inequality on Hook Lengths


 
 
 By reinterpreting the descent polynomial as a function enumerating standard Young tableaux of a ribbon shape, we use Naruse's hook-length formula to express the descent polynomial as a product of two polynomials: one is a trivial part which is a product of linear factors, and the other comes from the excitation factor of Naruse's formula. We expand the excitation factor positively in a Newton basis which arises naturally from Naruse's formula. Under this expansion, each coefficient is the weight of a certain combinatorial object, which we introduce in this paper. We introduce and prove the "Slice and Push Inequality", which compares the weights of such combinatorial objects. As a consequence, we establish a proof of a conjecture by Diaz-Lopez et al. that bounds the roots of descent polynomials.
 
 

ON $n$-NODE LINES IN $GC_n$ SETS

An $n$-poised node set $\mathcal X$ in the plane is called $GC_n$ set, if the fundamental polynomial of each node is a product of linear factors. A line is called $k$-node line, if it passes through exactly $k$-nodes of $\mathcal X.$ At most $n+1$ nodes can be collinear in $\Xset$ and an $(n+1)$-node line is called maximal line. The well-known conjecture of M. Gasca and J.I. Maeztu states that every $GC_n$ set has a maximal line. Until now the conjecture has been proved only for the cases $n \le 5.$ In this paper we prove some results concerning $n$-node lines, assuming that the Gasca--Maeztu conjecture is true.

Spectral invariants for finite dimensional Lie algebras

For a Lie algebra L with basis {x1,x2,…,xn}, its associated characteristic polynomial QL(z) is the determinant of the linear pencil z0I+z1adx1+⋯+znadxn. This paper shows that QL is invariant under the automorphism group Aut(L). The zero variety and factorization of QL reflect the structure of L. In the case L is solvable QL is known to be a product of linear factors. This fact gives rise to the definition of spectral matrix and the Poincaré polynomial for solvable Lie algebras. Application is given to 1-dimensional extensions of nilpotent Lie algebras.

Linear projections of the Vandermonde polynomial

An n-variate Vandermonde polynomial is the determinant of the n×n matrix where the ith column is the vector (1,xi,xi2,…,xin−1)T. Vandermonde polynomials play a crucial role in the theory of alternating polynomials and are useful in Lagrangian polynomial interpolation which arise in the theory of error correcting codes. In this work we study structural and computational aspects of linear projections of Vandermonde polynomials.First, we consider the problem of testing if a given polynomial is equivalent to the Vandermonde polynomial. We obtain a deterministic polynomial time algorithm to test if f is linearly equivalent to the Vandermonde polynomial when f is given as product of linear factors. In the case when f is given as a black-box our algorithm runs in randomized polynomial time.Exploring the structure of projections of Vandermonde polynomials further, we describe the group of symmetries of a Vandermonde polynomial and obtain a basis for the associated Lie algebra.Finally, we study arithmetic circuits built over projections of Vandermonde polynomials. We show universality property for some of the models and obtain a lower bounds against sum of projections of Vandermonde determinant.

Counting factorisations of monomials over rings of integers modulo N

A sharp bound is obtained for the number of ways to express the monomial X n as a product of linear factors over ℤ/p α ℤ. The proof relies on an induction-on-scale procedure which is used to estimate the number of solutions to a certain system of polynomial congruences. The method also applies to more general systems of polynomial congruences that satisfy a non-degeneracy hypothesis.

An algorithm which outputs a graph with a specified chromatic factor

For an undirected graph G, the chromatic polynomial is a key specialisation of its Tutte polynomial. The “(α+n)-conjecture” states that given any algebraic integer α, for large enough n∈N the shifted value α+n is the root of some chromatic polynomial. We develop an algorithm whose input is a monic polynomial Pd(X)∈Z[X] of degree d≤4, and the output is a (d,N)-biclique whose chromatic polynomial is a product of linear factors and the factor Pd(X−n) where n∈N is taken to be sufficiently large, provided that such a (d,N)-biclique exists. In degree d=3 our method tends to produce smaller (3,N)-bicliques than Adam Bohn’s cubic parameterisation (Bohn, 2014), while in degree d=4 it provides an efficient way to check the (α+n)-conjecture for a given quartic polynomial. The construction relies on previous work of Farrell (1980) which interprets chromatic coefficients via induced subgraphs, together with a simple search algorithm that is based on the number of ways to partition an integer. We also derive new expressions for the interesting factors of the chromatic polynomials for (3,N)-bicliques and (4,N)-bicliques, in terms of these induced subgraphs.

Extensions of planar GC sets and syzygy matrices

The geometric characterization, introduced by Chung and Yao, identifies node sets for total degree interpolation such that the Lagrange fundamental polynomials are products of linear factors. Sets satisfying the geometric characterization are usually called GC sets. Gasca and Maeztu conjectured that planar GC sets of degree n contain n + 1 collinear points. It has been shown that the conjecture holds for degrees not greater than 5 but it is still unsolved for general degree. One promising approach consists of studying the syzygies of the ideal of polynomials vanishing at the nodes. In order to describe syzygy matrices of GC sets, we analyze the extension of a GC set of degree n to a GC set of degree n + 1, by adding a n + 2 nodes on a line.

Algebraic Properties of Chromatic Roots

A chromatic root is a root of the chromatic polynomial of a graph. Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institute in late 2008. The purpose of this paper is to report on the seminar and subsequent developments.We conjecture that, for every algebraic integer $\alpha$, there is a natural number n such that $\alpha+n$ is a chromatic root. This is proved for quadratic integers; an extension to cubic integers has been found by Adam Bohn. The idea is to consider certain special classes of graphs for which the chromatic polynomial is a product of linear factors and one "interesting" factor of larger degree. We also report computational results on the Galois groups of irreducible factors of the chromatic polynomial for some special graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.

Laplace Inverse Transform for Functions of Type nth Root of a Product of Linear Factors

In this work, we present four results for the Laplace inverse transform of functions that involve the nth root of a product of linear factors. In order to find the Laplace inverse transform, we considered a branch cut for the nth root and a region of suitable integration, to avoid the branching points. Due to that, the solution is in terms of integrals, we easily approach this solution for some specific parameters.

ON A CONJECTURE IN BIVARIATE INTERPOLATION

Denote the space of all bivariate polynomials of total degree n$\leq n$ by $\Pi_n$. We are interested in n-poised sets of nodes with the property that the fundamental polynomial of each node is a product of linear factors. In 1981 M. Gasca and J. I.Maeztu conjectured that every such set contains necessarily $n+1$ collinear nodes. Up to now this had been confirmed for degrees $n \leq 5$. Here we bring a simple and short proof of the conjecture for $n = 4$.

Addendum to “Factoring skew polynomials over Hamilton's quaternion algebra and the complex numbers” [J. Algebra 427 (2015) 20–29

Let D be the quaternion division algebra over a real closed field F. Then every non-constant polynomial in a skew-polynomial ring D[t;σ,δ] decomposes into a product of linear factors, and thus has a zero in D. This improves [8, Theorem 2].

Factoring skew polynomials over Hamilton's quaternion algebra and the complex numbers

We show that all non-constant polynomials in a skew-polynomial ring H[t;σ,δ] over Hamilton's quaternions decompose into a product of linear factors, and that all non-constant polynomials in the skew-polynomial ring C[t;σ,δ] decompose into a product of linear and quadratic irreducible factors. Our proofs use nonassociative algebras constructed out of skew-polynomial rings as introduced by Petit.Applying results by Petit, we also characterize the real division algebras which are two-dimensional vector spaces over C with C in their left and middle nuclei.

The Gasca–Maeztu conjecture for $$n=5$$ n = 5

An $$n$$ n -poised set in two dimensions is a set of nodes admitting unique bivariate interpolation with polynomials of total degree at most $$n$$ n . We are interested in poised sets with the property that all fundamental polynomials are products of linear factors. Gasca and Maeztu (Numer Math 39:1---14, 1982) conjectured that every such set necessarily contains $$n+1$$ n + 1 collinear nodes. Up to now, this had been confirmed only for $$n\le 4$$ n ≤ 4 , the case $$n=4$$ n = 4 having been proved for the first time by Busch (Rev Un Mat Argent 36:33---38, 1990). In the present paper, we prove the case $$n=5$$ n = 5 with new methods that might also be useful in deciding the still open cases for $$n\ge 6$$ n ? 6 .

A multivariate hook formula for labelled trees

Several hook summation formulae for binary trees have appeared recently in the literature. In this paper we present an analogous formula for unordered increasing trees of size r, which involves r parameters. The right-hand side can be written nicely as a product of linear factors. We study two specializations of this new formula, including Cayleyʼs enumeration of trees with respect to vertex degree. We give three proofs of the hook formula. One of these proofs arises somewhat indirectly, from representation theory of the symmetric groups, and in particular uses Kerovʼs character polynomials. The other proofs are more direct, and of independent interest.

Akar-akar Polinomial Separable sebagai Pembentuk Perluasan Normal pada Ring Modulo

One of the most important uses of the ring and field theory is an extension of a broader field so that a polynomial can be found to have roots. In this study researchers took modulo a prima as follows indeterminate coeffcients to search for his roots extension the solutions of that it can seen normal. A field is subject to a polynomial form a set of polynomials , where is a coefficient field its terms modulo a prime number. Of the set of polynomial exists a polynomial is irreducible, it is necessary to extension the field to know the roots of the solution. Suppose to extension of the field is a field . Field is called extension the field over a field , if the field is subfield of the field and is irreducible polynomial in then can be factored as a product of linear factors in the splitting field. If the polynomial has different roots in the splitting field the polynomial is called polynomial separable. In this study polynomial separable is contained of odd degree in which the coefficients of the tribes polynomial is contained in the extension field. Polynomial is called a polynomial separable odd because it has different roots in the factors and there is one factor in a polynomial in the field. Splitting field that contains all the set of polynomials separable is called normal extension.

Bose and Fermi walk configurations on planar graphs

The number, fCn(H), of n-walk configurations of type C is investigated on certain two-rooted directed planar graphs H which will be always realized as plane graphs in R2. C may be Bose or Fermi as defined by Inui and Katori. Both types of configuration are collections of non-crossing walks which follow the directed paths between the roots of the plane graph H. In the case of configurations of Fermi type each walk may be included only once. The number fBosen(H) is shown to be a polynomial in n of degree nmax − 1 where nmax is the maximum number of walks in a Fermi configuration. The coefficient of the highest power of n in this polynomial is simply related to the number of maximal Fermi walk configurations. It is also shown that nmax = c(H) + 1 where c(H) is the number of finite faces on H. Extension of these results to multi-rooted graphs is also discussed. When H is the union of paths between two sites of the directed square lattice subject to various boundary conditions Kreweras showed that the number of Bose configurations is equal to the number of n-element multi-chains on segments of Young’s lattice. He expressed this number as a determinant the elements of which are polynomials in n. We evaluate this determinant by the method of LU decomposition in the case of ‘watermelon’ configurations above a wall. In this case the polynomial is a product of linear factors but on introducing a second wall the polynomial does not completely factorize but has a factor which is the number of watermelon configurations on the largest rectangular subgraph. The number of two-rooted ‘star’ configurations is found to be the product of the numbers of watermelon configurations on the three rectangular subgraphs into which it may be partitioned.

Generation of the symmetric field by Newton polynomials in prime characteristic

Let $N_m = x^m + y^m$ be the $m$-th Newton polynomial in two variables, for $m \geq 1$. Dvornicich and Zannier proved that in characteristic zero three Newton polynomials $N_a, N_b, N_c$ are always sufficient to generate the symmetric field in $x$ and $y$, provided that $a,b,c$ are distinct positive integers such that $(a,b,c)=1$. In the present paper we prove that in case of prime characteristic $p$ the result still holds, if we assume additionally that $a,b,c,a-b,a-c,b-c$ are prime with $p$. We also provide a counterexample in the case where one of the hypotheses is missing. The result follows from the study of the factorization of a generalized Vandermonde determinant in three variables, that under general hypotheses factors as the product of a trivial Vandermonde factor and an irreducible factor. On the other side, the counterexample is connected to certain cases where the Schur polynomials factor as a product of linear factors.

Decomposability of matrix polynomials with commuting coefficients into a product of linear factors

The list of known sets of factorizable matrix polynomials is supplemented by new sets of polynomials of this sort. The known set of nonfactorizable matrix polynomials is extended. These results can be applied to the study of polynomial equations and systems of differential equations with constant coefficients.

Punctured combinatorial Nullstellensätze

In this article we extend Alon’s Nullstellensatz to functions which have multiple zeros at the common zeros of some polynomials g 1,g 2, …, g n , that are the product of linear factors. We then prove a punctured version which states, for simple zeros, that if f vanishes at nearly all, but not all, of the common zeros of g 1(X 1), …,g n (X n ) then every residue of f modulo the ideal generated by g 1, …, g n , has a large degree. This punctured Nullstellensatz is used to prove a blocking theorem for projective and affine geometries over an arbitrary field. This theorem has as corollaries a theorem of Alon and Furedi which gives a lower bound on the number of hyperplanes needed to cover all but one of the points of a hypercube and theorems of Bruen, Jamison and Brouwer and Schrijver which provides lower bounds on the number of points needed to block the hyperplanes of an affine space over a finite field.

On the power-free values of polynomials in two variables: II

The following theorem is proved. Suppose that for integers r , s ⩾ 2 , f ( x , y ) is an inhomogeneous polynomial of degree r with rational integral coefficients that is irreducible over the rationals, that is not a polynomial in a linear combination of x and y, that has no fixed sth power divisors other than 1, and that is a product of linear factors over some extension field of the rationals. Then, if N ( X ) denote the number of integers m , n of magnitude not exceeding X for which f ( m , n ) is sth power-free, the asymptotic formula N ( X ) ∼ 4 X 2 ∏ p ( 1 − ρ ( p s ) p 2 s ) is valid for s ⩾ 1 2 r − 1 , where ρ ( l ) is the number of incongruent zeros, mod l, of f ( x , y ) . In these circumstances f ( m , n ) is infinitely often sth power-free. This result improves upon a lower bound found for N ( X ) in a previous paper [C. Hooley, On the power-free values of polynomials in two variables, Roth 80th birthday volume, in press].