On the Real-Rootedness of  the Local $h$-Polynomials of Edgewise Subdivisions

Zhang, Philip B.

doi:10.37236/7492

Cited by 7 publications

(6 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, [3,Question 4.11] further asked if the γ-nonnegativity of each of these local h-polynomials follows from real-rootedness. In [49] real-rootedness of d n (z), the local h-polynomial of the barycentric subdivision of a simplex, is verified, and more recently in [30,50] the real-rootedness of the local h-polynomial of the r th edgewise subdivision of a simplex was shown. The results in Sections 3 and 4 provide an alternative proof for these results, and also allow us to verify [3, Question 4.11] for the final missing example.…”

Section: Applicationsmentioning

confidence: 99%

Derangements, Ehrhart theory, and local h-polynomials

Gustafsson¹,

Solus²

2020

Advances in Mathematics

View full text Add to dashboard Cite

The Eulerian polynomials and derangement polynomials are two well-studied generating functions that frequently arise in combinatorics, algebra, and geometry. When one makes an appearance, the other often does so as well, and their corresponding generalizations are similarly linked. This is this case in the theory of subdivisions of simplicial complexes, where the Eulerian polynomial is an h-polynomial and the derangement polynomial is its local h-polynomial. Separately, in Ehrhart theory the Eulerian polynomials are generalized by the h * -polynomials of s-lecture hall simplices. Here, we show that derangement polynomials are analogously generalized by the box polynomials, or local h * -polynomials, of the s-lecture hall simplices, and that these polynomials are all real-rooted. We then connect the two theories by showing that the local h-polynomials of common subdivisions in algebra and topology are realized as local h * -polynomials of s-lecture hall simplices. We use this connection to address some open questions on real-rootedness and unimodality of generating polynomials, some from each side of the story.

show abstract

Section: Applicationsmentioning

confidence: 99%

Derangements, Ehrhart theory, and local h-polynomials

Gustafsson¹,

Solus²

2020

Advances in Mathematics

View full text Add to dashboard Cite

show abstract

“…All the matrices on the right hand side preserve interlacing, which has already been checked in [21,33]. So do the matrices on the left hand side.…”

Section: Interlacingmentioning

confidence: 65%

Real-rootedness of variations of Eulerian polynomials

Haglund

Zhang

2019

Advances in Applied Mathematics

Self Cite

View full text Add to dashboard Cite

The binomial Eulerian polynomials, introduced by Postnikov, Reiner, and Williams, are γ-positive polynomials and can be interpreted as h-polynomials of certain flag simplicial polytopes. Recently, Athanasiadis studied analogs of these polynomials for colored permutations. In this paper, we generalize them to s-inversion sequences and prove that these new polynomials have only real roots by the method of interlacing polynomials. Three applications of this result are presented. The first one is to prove the real-rootedness of binomial Eulerian polynomials, which confirms a conjecture of Ma, Ma, and Yeh. The second one is to prove that the symmetric decomposition of binomial Eulerian polynomials for colored permutations is real-rooted. Thirdly, our polynomials for certain s-inversion sequences are shown to admit a similar geometric interpretation related to edgewise subdivisions of simplexes.

show abstract

“…Despite strong interest, there were very few results about Conjectures 1.2 and 1.3. For any n ≥ 1, it was known that Conjectures 1.2 and 1.3 are both valid for the uniform matroid of rank n − 1 on n elements, see [12,32]. Based on the theory of multiplier sequences and n-sequences (see [5,6]), we prove that Conjecture 1.2 holds for fan matroids, wheel matroids and whirl matroids, and Conjecture 1.3 holds for fan matroids.…”

Section: Introductionmentioning

confidence: 79%

Kazhdan-Lusztig polynomials of fan matroids, wheel matroids and whirl matroids

Lü¹,

Xie²,

Yang³

2018

Preprint

View full text Add to dashboard Cite

The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias, Proudfoot and Wakefield, whose properties need to be further explored. In this paper we prove that the Kazhdan-Lusztig polynomials of fan matroids coincide with Motzkin polynomials, which was recently conjectured by Gedeon. As a byproduct, we determine the Kazhdan-Lusztig polynomials of graphic matroids of squares of paths. We further obtain explicit formulas of the Kazhdan-Lusztig polynomials of wheel matroids and whirl matroids. We prove the real-rootedness of the Kazhdan-Lusztig polynomials of these matroids, which provides positive evidence for a conjecture due to Gedeon, Proudfoot and Young. Based on the results on the Kazhdan-Lusztig polynomials, we also determine the Z-polynomials of fan matroids, wheel matroids and whirl matroids, and prove their real-rootedness, which provides further evidence in support of a conjecture of Proudfoot, Xu, and Young.

show abstract

On the Real-Rootedness of the Local $h$-Polynomials of Edgewise Subdivisions

Cited by 7 publications

References 17 publications

Derangements, Ehrhart theory, and local h-polynomials

Derangements, Ehrhart theory, and local h-polynomials

Real-rootedness of variations of Eulerian polynomials

Kazhdan-Lusztig polynomials of fan matroids, wheel matroids and whirl matroids

Contact Info

Product

Resources

About