Polynomials with and without determinantal representations

The problem of writing real zero polynomials as determinants of linear matrix polynomials has recently attracted a lot of attention. Helton and Vinnikov [9] have proved that any real zero polynomial in two variables has a determinantal representation. Brändén [2] has shown that the result does not extend to arbitrary numbers of variables, disproving the generalized Lax conjecture. We prove that in fact almost no real zero polynomial admits a determinantal representation; there are dimensional differences between the two sets. The result follows from a general upper bound on the size of linear matrix polynomials. We then provide a large class of surprisingly simple explicit real zero polynomials that do not have a determinantal representation. We finally characterize polynomials of which some power has a determinantal representation, in terms of an algebra with involution having a finite dimensional representation. We use the characterization to prove that any quadratic real zero polynomial has a determinantal representation, after taking a high enough power. Taking powers is thereby really necessary in general. The representations emerge explicitly, and we characterize them up to unitary equivalence.     

Stable multivariate W-Eulerian polynomials

We prove a multivariate strengthening of Brenti?s result that every root of the Eulerian polynomial of type B is real. Our proof combines a refinement of the descent statistic for signed permutations with the notion of real stability—a generalization of real-rootedness to polynomials in multiple variables. The key is that our refined multivariate Eulerian polynomials satisfy a recurrence given by a stability-preserving linear operator.Our results extend naturally to colored permutations, and we also give stable generalizations of recent real-rootedness results due to Dilks, Petersen, and Stembridge on affine Eulerian polynomials of types A and C. Finally, although we are not able to settle Brenti?s real-rootedness conjecture for Eulerian polynomials of type D, nor prove a companion conjecture of Dilks, Petersen, and Stembridge for affine Eulerian polynomials of types B and D, we indicate some methods of attack and pose some related open problems.     

Schur positivity and the <Emphasis Type="Italic">q</Emphasis>-log-convexity of the Narayana polynomials

We prove two recent conjectures of Liu and Wang by establishing the strong q-log-convexity of the Narayana polynomials, and showing that the Narayana transformation preserves log-convexity. We begin with a formula of Brändén expressing the q-Narayana numbers as a specialization of Schur functions and, by deriving several symmetric function identities, we obtain the necessary Schur-positivity results. In addition, we prove the strong q-log-concavity of the q-Narayana numbers. The q-log-concavity of the q-Narayana numbers N _q (n,k) for fixed k is a special case of a conjecture of McNamara and Sagan on the infinite q-log-concavity of the Gaussian coefficients.     

Stable multivariate Eulerian polynomials and generalized Stirling permutations

We study Eulerian polynomials as the generating polynomials of the descent statistic over Stirling permutations—a class of restricted multiset permutations. We develop their multivariate refinements by indexing variables by the values at the descent tops, rather than the position where they appear. We prove that the obtained multivariate polynomials are stable, in the sense that they do not vanish whenever all the variables lie in the open upper half-plane. Our multivariate construction generalizes the multivariate Eulerian polynomial for permutations, and extends naturally to r-Stirling and generalized Stirling permutations.The benefit of this refinement is manifold. First of all, the stability of the multivariate generating functions implies that their univariate counterparts, obtained by diagonalization, have only real roots. Second, we obtain simpler recurrences of a general pattern, which allows for essentially a single proof of stability for all the cases, and further proofs of equidistributions among different statistics. Our approach provides a unifying framework of some recent results of Bóna, Brändén, Brenti, Janson, Kuba, and Panholzer. We conclude by posing several interesting open problems.     

Introduction to the combinatorial atlas

We give elementary self-contained proofs of the strong Mason conjecture recently proved by Anari et al. (2018) and Brändén and Huh (2020), and of the classical Alexandrov–Fenchel inequality. Both proofs use the combinatorial atlas technology recently introduced by the authors Chan and Pak (2021). We also give a formal relationship between combinatorial atlases and Lorentzian polynomials.     

Strong factorization property of Macdonald polynomials and higher-order Macdonald’s positivity conjecture

We prove a strong factorization property of interpolation Macdonald polynomials when q tends to 1. As a consequence, we show that Macdonald polynomials have a strong factorization property when q tends to 1, which was posed as an open question in our previous paper with Féray. Furthermore, we introduce multivariate q, t-Kostka numbers and we show that they are polynomials in q, t with integer coefficients by using the strong factorization property of Macdonald polynomials. We conjecture that multivariate q, t-Kostka numbers are in fact polynomials in q, t with nonnegative integer coefficients, which generalizes the celebrated Macdonald’s positivity conjecture.     

The symmetric and unimodal expansion of Eulerian polynomials via continued fractions

This paper was motivated by a conjecture of Brändén [P. Brändén, Actions on permutations and unimodality of descent polynomials, European J. Combin. 29 (2) (2008) 514-531] about the divisibility of the coefficients in an expansion of generalized Eulerian polynomials, which implies the symmetric and unimodal property of the Eulerian numbers. We show that such a formula with the conjectured property can be derived from the combinatorial theory of continued fractions. We also discuss an analogous expansion for the corresponding formula for derangements and prove a (p,q)-analogue of the fact that the (-1)-evaluation of the enumerator polynomials of permutations (resp. derangements) by the number of excedances gives rise to tangent numbers (resp. secant numbers). The (p,q)-analogue unifies and generalizes our recent results [H. Shin, J. Zeng, The q-tangent and q-secant numbers via continued fractions, European J. Combin. 31 (7) (2010) 1689-1705] and that of Josuat-Vergès [M. Josuat-Vergés, A q-enumeration of alternating permutations, European J. Combin. 31 (7) (2010) 1892-1906].     

Two new triangles of q-integers via q-Eulerian polynomials of type A and B

The classical Eulerian polynomials can be expanded in the basis t ^k?1(1+t) ^n+1?2k (1≤k≤?(n+1)/2?) with positive integral coefficients. This formula implies both the symmetry and the unimodality of the Eulerian polynomials. In this paper, we prove a q-analogue of this expansion for Carlitz’s q-Eulerian polynomials as well as a similar formula for Chow–Gessel’s q-Eulerian polynomials of type B. We shall give some applications of these two formulas, which involve two new sequences of polynomials in the variable q with positive integral coefficients. It is an open problem to give a combinatorial interpretation for these polynomials.     

EXOTIC SYMMETRIC SPACE OVER A FINITE FIELD,II

This paper is the second of the papers of the same title. In this paper, we prove a conjecture of Achar–Henderson, which asserts that the Poincaré polynomials of the intersection cohomology complex associated to the closure of Sp_2n -orbits in the Kato's exotic nilpotent cone coincide with the modified Kostka polynomials indexed by double partitions, introduced by the first author. Actually, this conjecture was recently proved by Kato by a different method. Our approach is based on the theory of character sheaves on the exotic symmetric space.     

Log-convex and Stieltjes moment sequences

We show that Stieltjes moment sequences are infinitely log-convex, which parallels a famous result that (finite) Pólya frequency sequences are infinitely log-concave. We introduce the concept of q-Stieltjes moment sequences of polynomials and show that many well-known polynomials in combinatorics are such sequences. We provide a criterion for linear transformations and convolutions preserving Stieltjes moment sequences. Many well-known combinatorial sequences are shown to be Stieltjes moment sequences in a unified approach and therefore infinitely log-convex, which in particular settles a conjecture of Chen and Xia about the infinite log-convexity of the Schröder numbers. We also list some interesting problems and conjectures about the log-convexity and the Stieltjes moment property of the (generalized) Apéry numbers.     

The No Gap Conjecture for tame hereditary algebras

The “No Gap Conjecture” of Brüstle–Dupont–Pérotin states that the set of lengths of maximal green sequences for hereditary algebras over an algebraically closed field has no gaps. This follows from a stronger conjecture that any two maximal green sequences can be “polygonally deformed” into each other. We prove this stronger conjecture for all tame hereditary algebras over any field, equivalently, for any acyclic tame skew-symmetrizable exchange matrix.     

Ensembles de Julia quadratiques de mesure de Lebesgue strictement positive

In this Note, we prove a variant of a conjecture stated in the thesis of Chéritat. The proof is based on results announced by Inou and Shishikura, and on earlier results of McMullen and of Chéritat. According to Chéritat's thesis, this allows us to complete a plan initiated by Douady and to show that there exist quadratic polynomials having a Julia set of positive Lebesgue measure. To cite this article: X. Buff, A. Chéritat, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Multivariate splines and polytopes

In this paper, we use multivariate splines to investigate the volume of polytopes. We first present an explicit formula for the multivariate truncated power, which can be considered as a dual version of the famous Brion’s formula for the volume of polytopes. We also prove that the integration of polynomials over polytopes can be dealt with by using the multivariate truncated power. Moreover, we show that the volume of cube slicing can be considered as the maximum value of the box spline. On the basis of this connection, we give a simple proof for Good’s conjecture, which has been settled before by probability methods.     

A Generalization of a theorem of chang

Szigeti, Tuza and Révész have developed a method in [6] to obtain polynomial identities for the n×n matrix ring over a commutative ring starting from directed Eulerian graphs. These polynomials are called Euler-ian. In the first part of this paper we show some polynomials that are in the T-ideal generated by a certain set of Eulerian polynomials, hence we get some identities of the n×n matrices. This result is a generalization of a theorem of Chang [l]. After that, using this theorem, we show that any Eulerian identity arising from a graph which lias d-fold multiple edges follows from the standard identity of degree d     

Euler polynomials,Bernoulli polynomials,and Lévyʼs stochastic area formula

In 1951, P. Lévy represented the Euler and Bernoulli numbers in terms of the moments of Lévy?s stochastic area. Recently the authors extended his result to the case of Eulerian polynomials of types A and B. In this paper, we continue to apply the same method to the Euler and Bernoulli polynomials, and will express these polynomials with the use of Lévy?s stochastic area. Moreover, a natural problem, arising from such representations, to calculate the expectations of polynomials of the stochastic area and the norm of the Brownian motion will be solved.     

The Mukai conjecture for log Fano manifolds

For a log Fano manifold (X,D) with D ≠ 0 and of the log Fano pseudoindex ≥2, we prove that the restriction homomorphism Pic(X) → Pic(D ₁) of Picard groups is injective for any irreducible component D ₁ ? D. The strategy of our proof is to run a certain minimal model program and is similar to Casagrande’s argument. As a corollary, we prove that the Mukai conjecture (resp. the generalized Mukai conjecture) implies the log Mukai conjecture (resp. the log generalized Mukai conjecture).     

Bivariate Gonarov polynomials and integer sequences

Univariate Gonarov polynomials arose from the Gonarov interpolation problem in numerical analysis.They provide a natural basis of polynomials for working with u-parking functions,which are integer sequences whose order statistics are bounded by a given sequence u.In this paper,we study multivariate Gonarov polynomials,which form a basis of solutions for multivariate Gonarov interpolation problem.We present algebraic and analytic properties of multivariate Gonarov polynomials and establish a combinatorial relation with integer sequences.Explicitly,we prove that multivariate Gonarov polynomials enumerate k-tuples of integers sequences whose order statistics are bounded by certain weights along lattice paths in Nk.It leads to a higher-dimensional generalization of parking functions,for which many enumerative results can be derived from the theory of multivariate Gonarov polynomials.     

On Multiple and Infinite Log-Concavity

Following Boros-Moll, a sequence (a _n ) is m-log-concave if \({\mathcal{L}^{j}(a_{n})\geqslant0}\) for all j =  0, 1, . . . , m. Here, \({\mathcal{L}}\) is the operator defined by \({\mathcal{L}(a_{n}) = a^{2}_{n}-a_{n-1}a_{n+1}}\). By a criterion of Craven-Csordas and McNamara-Sagan it is known that a sequence is ∞-log-concave if it satisfies the stronger inequality \({a^{2}_{k}\geqslant ra_{k-1}a_{k+1}}\) for large enough r. On the other hand, a recent result of Brändén shows that ∞-log-concave sequences include sequences whose generating polynomial has only negative real roots. In this paper, we investigate sequences which are fixed by a power of the operator \({\mathcal{L}}\) and are therefore ∞-log-concave for a very different reason. Surprisingly, we find that sequences fixed by the non-linear operators \({\mathcal{L}}\) and \({\mathcal{L}^{2}}\) are, in fact, characterized by a linear 4-term recurrence. In a final conjectural part, we observe that positive sequences appear to become ∞-log-concave if convoluted with themselves a finite number of times.     

Milnor fibration at infinity for mixed polynomials

We study the existence of Milnor fibration on a big enough sphere at infinity for a mixed polynomial f: ?²ⁿ → ?². By using strongly non-degenerate condition, we prove a counterpart of Némethi and Zaharia’s fibration theorem. In particular, we obtain a global version of Oka’s fibration theorem for strongly non-degenerate and convenient mixed polynomials.     

On the distribution of algebraic primes in small regions

Assuming a uniform version of Gross and Smith’s generalized k-tuple conjecture, we prove that the prime elements in a ring of algebraic integers have Poisson distribution in small regions, by considering the sum of the corresponding singular series.