On some classes of polynomials with nonnegative coefficients and a given factor

For a given real polynomial f without positive roots we study polynomials g of lowest degree such that the product gf has positive (nonnegative, respectively) coefficients. We show that for quadratic f with negative linear coefficient every such g must have positive coefficients and exhibit an easy procedure for the determination of g. If f has only integer coefficients we show that g with integer coefficients can be found. Furthermore, for some classes of polynomials f we give upper (lower, respectively) bounds for the degrees of g.     

Gale duality bounds for roots of polynomials with nonnegative coefficients

We bound the location of roots of polynomials that have nonnegative coefficients with respect to a fixed but arbitrary basis of the vector space of polynomials of degree at most d. For this, we interpret the basis polynomials as vector fields in the real plane, and at each point in the plane analyze the combinatorics of the Gale dual vector configuration. This approach permits us to incorporate arbitrary linear equations and inequalities among the coefficients in a unified manner to obtain more precise bounds on the location of roots. We apply our technique to bound the location of roots of Ehrhart and chromatic polynomials. Finally, we give an explanation for the clustering seen in plots of roots of random polynomials.     

On realizability of sign patterns by real polynomials

The classical Descartes’ rule of signs limits the number of positive roots of a real polynomial in one variable by the number of sign changes in the sequence of its coefficients. One can ask the question which pairs of nonnegative integers (p, n), chosen in accordance with this rule and with some other natural conditions, can be the pairs of numbers of positive and negative roots of a real polynomial with prescribed signs of the coefficients. The paper solves this problem for degree 8 polynomials.     

On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series

For every positive integer n, consider the linear operator U _n on polynomials of degree at most d with integer coefficients defined as follows: if we write ${\frac{h(t)}{(1 - t)^{d + 1}}=\sum_{m \geq 0} g(m) \, t^{m}}For every positive integer n, consider the linear operator U _n on polynomials of degree at most d with integer coefficients defined as follows: if we write \frach(t)(1 - t)^{d + 1}=?_{m 3 0} g(m)  t^m{\frac{h(t)}{(1 - t)^{d + 1}}=\sum_{m \geq 0} g(m) \, t^{m}} , for some polynomial g(m) with rational coefficients, then \fracU_nh(t)(1- t)^d+1 = ?_{m 3 0}g(nm)  t^m{\frac{{\rm{U}}_{n}h(t)}{(1- t)^{d+1}} = \sum_{m \geq 0}g(nm) \, t^{m}} . We show that there exists a positive integer n _d , depending only on d, such that if h(t) is a polynomial of degree at most d with nonnegative integer coefficients and h(0) = 1, then for n ≥ n _d , U _n h(t) has simple, real, negative roots and positive, strictly log concave and strictly unimodal coefficients. Applications are given to Ehrhart δ-polynomials and unimodular triangulations of dilations of lattice polytopes, as well as Hilbert series of Veronese subrings of Cohen–Macauley graded rings.     

On the f- and h-triangle of the barycentric subdivision of a simplicial complex

For a simplicial complex Δ we study the behavior of its f- and h-triangle under the action of barycentric subdivision. In particular we describe the f- and h-triangle of its barycentric subdivision sd(Δ). The same has been done for f- and h-vector of sd(Δ) by F. Brenti, V. Welker (2008). As a consequence we show that if the entries of the h-triangle of Δ are nonnegative, then the entries of the h-triangle of sd(Δ) are also nonnegative. We conclude with a few properties of the h-triangle of sd(Δ).     

Three positive solutions of semilinear elliptic equations in exterior cylinder domains

In this paper, assume that h is nonnegative and ‖hL²_‖>0, we prove that if ‖hL²_‖ is sufficiently small, then there are at least three positive solutions of Eq. (1) in an exterior cylinder domain.     

Structure of the zeros ofq-Bernoulli polynomials

In this paper we observe the structure of the roots ofq-Bernoulli polynomials,β _n (w, h|q), using numerical investigation. By numerical experiments, we demonstrate a remarkably regular structure of the real roots ofβ _n (w, h|q) forq=?1/5, ?1/2. Finally, we give a table for numbers of real and complex zeros ofβ _n (w, h|q).     

Coxeter cones and their h-vectors

Coxeter cones are formed by intersecting the nonnegative sides of a collection of root hyperplanes in some root system. They are shellable subcomplexes of the Coxeter complex, and their h-vectors record the distribution of descents among their chambers. We identify a natural class of “graded” Coxeter cones with the property that their h-vectors are symmetric and unimodal, thereby generalizing recent theorems of Reiner-Welker and Brändén about the Eulerian polynomials of graded partially ordered sets.     

An extremal problem for algebraic polynomials with zero mean value on an interval

LetP _n ^O(h) be the set of algebraic polynomials of degreen with real coefficients and with zero mean value (with weighth) on the interval [?1, 1]: $$\smallint _{ - 1}^1 h(x)p_n (x) dx = 0;$$ hereh is a function which is summable, nonnegative, and nonzero on a set of positive measure on [?1, 1]. We study the problem of the least possible value $$i_n (h) = \inf \{ \mu (p_n ):p_n \in \mathcal{P}_n^0 \} $$ of the measure μ(P _n)=mes{x∈[?1,1]:P _n(x)≥0} of the set of points of the interval at which the polynomialp _n∈P _n ^O is nonnegative. We find the exact value ofi _n(h) under certain restrictions on the weighth. In particular, the Jacobi weight $$h^{(\alpha ,\beta )} (x) = (1 - x)^\alpha (1 + x)^\beta $$ satisfies these restrictions provided that ?1<α, β≤0.     

Root statistics of random polynomials with bounded Mahler measure

The Mahler measure of a polynomial is a measure of complexity formed by taking the modulus of the leading coefficient times the modulus of the product of its roots outside the unit circle. The roots of a real degree N polynomial chosen uniformly from the set of polynomials of Mahler measure at most 1 yield a Pfaffian point process on the complex plane. When N is large, with probability tending to 1, the roots tend to the unit circle, and we investigate the asymptotics of the scaled kernel in a neighborhood of a point on the unit circle. When this point is away from the real axis (on which there is a positive probability of finding a root) the scaled process degenerates to a determinantal point process with the same local statistics (i.e.   scalar kernel) as the limiting process formed from the roots of complex polynomials chosen uniformly from the set of polynomials of Mahler measure at most 1. Three new matrix kernels appear in a neighborhood of ±1 which encode information about the correlations between real roots, between complex roots and between real and complex roots. Away from the unit circle, the kernels converge to new limiting kernels, which imply among other things that the expected number of roots in any open subset of C$\mathbb{C}$ disjoint from the unit circle converges to a positive number. We also give ensembles with identical statistics drawn from two-dimensional electrostatics with potential theoretic weights, and normal matrices chosen with regard to their topological entropy as actions on Euclidean space.     

Factorization of cp-rank-3 completely positive matrices

A symmetric positive semi-definite matrix A is called completely positive if there exists a matrix B with nonnegative entries such that A = BB^?. If B is such a matrix with a minimal number p of columns, then p is called the cp-rank of A. In this paper we develop a finite and exact algorithm to factorize any matrix A of cp-rank 3. Failure of this algorithm implies that A does not have cp-rank 3. Our motivation stems from the question if there exist three nonnegative polynomials of degree at most four that vanish at the boundary of an interval and are orthonormal with respect to a certain inner product.     

Intersective polynomials and the primes

Intersective polynomials are polynomials in Z[x] having roots every modulus. For example, P₁(n)=n² and P₂(n)=n²−1 are intersective polynomials, but P₃(n)=n²+1 is not. The purpose of this note is to deduce, using results of Green and Tao (2006) [8] and Lucier (2006) [16], that for any intersective polynomial h, inside any subset of positive relative density of the primes, we can find distinct primes p₁,p₂ such that p₁−p₂=h(n) for some integer n. Such a conclusion also holds in the Chen primes (where by a Chen prime we mean a prime number p such that p+2 is the product of at most 2 primes).     

Positive solutions of the p-Laplacian involving a superlinear nonlinearity with zeros

Using a combination of several methods, such as variational methods, the sub and supersolutions method, comparison principles and a priori estimates, we study existence, multiplicity, and the behavior with respect to λ of positive solutions of p-Laplace equations of the form −Δpu=λh(x,u), where the nonlinear term has p-superlinear growth at infinity, is nonnegative, and satisfies h(x,a(x))=0 for a suitable positive function a. In order to manage the asymptotic behavior of the solutions we extend a result due to Redheffer and we establish a new Liouville-type theorem for the p-Laplacian operator, where the nonlinearity involved is superlinear, nonnegative, and has positive zeros.     

Hoffman polynomials of nonnegative irreducible matrices and strongly connected digraphs

For a nonnegative n × n matrix A, we find that there is a polynomial f(x)∈R[x] such that f(A) is a positive matrix of rank one if and only if A is irreducible. Furthermore, we show that the lowest degree such polynomial f(x) with tr f(A) = n is unique. Thus, generalizing the well-known definition of the Hoffman polynomial of a strongly connected regular digraph, for any irreducible nonnegative n × n matrix A, we are led to define its Hoffman polynomial to be the polynomial f(x) of minimum degree satisfying that f(A) is positive and has rank 1 and trace n. The Hoffman polynomial of a strongly connected digraph is defined to be the Hoffman polynomial of its adjacency matrix. We collect in this paper some basic results and open problems related to the concept of Hoffman polynomials.     

Moment Approximations for Set-Semidefinite Polynomials

The set of polynomials that are nonnegative over a subset of the nonnegative orthant (we call them set-semidefinite) have many uses in optimization. A common example of this type set is the set of copositive matrices, where we are effectively considering nonnegativity over the entire nonnegative orthant and are restricted to homogeneous polynomials of degree two. Lasserre (SIAM J. Optim., 21(3):864–885, 2011) has previously considered a method using moments in order to provide an outer approximation to this set, for nonnegativity over a general subset of the real space. In this paper, we shall show that, in the special case of considering nonnegativity over a subset of the nonnegative orthant, we can provide a new outer approximation hierarchy. This is based on restricting moment matrices to be completely positive, and it is at least as good as Lasserre’s method. This can then be relaxed to give tractable approximations that are still at least as good as Lasserre’s method. In doing this, we also provide interesting new insights into the use of moments in constructing these approximations.     

The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [−1,1]

We consider polynomials that are orthogonal on [−1,1] with respect to a modified Jacobi weight (1−x)^α(1+x)^βh(x), with α,β>−1 and h real analytic and strictly positive on [−1,1]. We obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval [−1,1], for the recurrence coefficients and for the leading coefficients of the orthonormal polynomials. We also deduce asymptotic behavior for the Hankel determinants and for the monic orthogonal polynomials on the interval [−1,1]. For the asymptotic analysis we use the steepest descent technique for Riemann-Hilbert problems developed by Deift and Zhou, and applied to orthogonal polynomials on the real line by Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou. In the steepest descent method we will use the Szeg? function associated with the weight and for the local analysis around the endpoints ±1 we use Bessel functions of appropriate order, whereas Deift et al. use Airy functions.     

Factorizations of tropical and sign polynomials

In this text, we study factorizations of polynomials over the tropical hyperfield and the sign hyperfield, which we call tropical polynomials and sign polynomials, respectively. We classify all irreducible polynomials in either case. We show that tropical polynomials factor uniquely into irreducible factors, but that unique factorization fails for sign polynomials. We describe division algorithms for tropical and sign polynomials by linear terms that correspond to roots of the polynomials.     

Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams

We consider the problem of finding the number of matrices over a finite field with a certain rank and with support that avoids a subset of the entries. These matrices are a q-analogue of permutations with restricted positions (i.e., rook placements). For general sets of entries, these numbers of matrices are not polynomials in q (Stembridge in Ann. Comb. 2(4):365, 1998); however, when the set of entries is a Young diagram, the numbers, up to a power of q?1, are polynomials with nonnegative coefficients (Haglund in Adv. Appl. Math. 20(4):450, 1998). In this paper, we give a number of conditions under which these numbers are polynomials in q, or even polynomials with nonnegative integer coefficients. We extend Haglund’s result to complements of skew Young diagrams, and we apply this result to the case where the set of entries is the Rothe diagram of a permutation. In particular, we give a necessary and sufficient condition on the permutation for its Rothe diagram to be the complement of a skew Young diagram up to rearrangement of rows and columns. We end by giving conjectures connecting invertible matrices whose support avoids a Rothe diagram and Poincaré polynomials of the strong Bruhat order.     

Optimization Over the Boolean Hypercube Via Sums of Nonnegative Circuit Polynomials

Various key problems from theoretical computer science can be expressed as polynomial optimization problems over the boolean hypercube. One particularly successful way to prove complexity bounds for these types of problems is based on sums of squares (SOS) as nonnegativity certificates. In this article, we initiate optimization problems over the boolean hypercube via a recent, alternative certificate called sums of nonnegative circuit polynomials (SONC). We show that key results for SOS-based certificates remain valid: First, for polynomials, which are nonnegative over the n-variate boolean hypercube with constraints of degree d there exists a SONC certificate of degree at most \(n+d\). Second, if there exists a degree d SONC certificate for nonnegativity of a polynomial over the boolean hypercube, then there also exists a short degree d SONC certificate that includes at most \(n^{O(d)}\) nonnegative circuit polynomials. Moreover, we prove that, in opposite to SOS, the SONC cone is not closed under taking affine transformation of variables and that for SONC there does not exist an equivalent to Putinar’s Positivstellensatz for SOS. We discuss these results from both the algebraic and the optimization perspective.

     

Construction of Normalized B-Splines for a Family of Smooth Spline Spaces Over Powell–Sabin Triangulations

We construct a suitable B-spline representation for a family of bivariate spline functions with smoothness r≥1 and polynomial degree 3r?1. They are defined on a triangulation with Powell–Sabin refinement. The basis functions have a local support, they are nonnegative, and they form a partition of unity. The construction involves the determination of triangles that must contain a specific set of points. We further consider a number of CAGD applications. We show how to define control points and control polynomials (of degree 2r?1), and we provide an efficient and stable computation of the Bernstein–Bézier form of such splines.