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Let B be an integral domain over a field K of characteristic 0. The derivation δ of B[Yd] = B[y1, . . . ,yd] is elementary if δ (B) = 0 and δ (yi) ∈ B, i = 1, . . . ,d. Then for d ≥ 2 the elements uij = δ (yi)yj-δ (yj)yi, 1 ≤ i < j ≤ d, belong to the algebra B[Yd]δ of constants of δ, and it is a natural question whether the B-algebra B[Yd]δ is generated by all uij. In this paper we consider the special case of B = K[Xd] = K[x1, . . . ,xd]. If δ (yi) = xi, i = 1, . . . ,d, this is the Nowicki conjecture from 1994, which was confirmed in several papers based on different methods. The case δ (yi) = xini, ni > 0, i = 1, . . . ,d, was handled by Khoury in the first proof of the Nowicki conjecture given by him in 2004. As a consequence of the proof of Kuroda in 2009, if δ (yi) = ƒi(xi), for any nonconstant polynomials ƒi(xi), i = 1, . . . ,d, then B[Yd]δ = K[Xd,Yd]δ is generated by Xd and Ud = uij = ƒi(xi)yj-yiƒj(xj) mid 1 ≤ i < j ≤ d. In the present paper we have found a presentation of the algebra K[Xd,Yd]δ = K[Xd,Ud mid R = S = 0],d ≥ 4, R = r(i,j,k,l)mid 1 ≥ i < j < k < l \≤ d, S = s(i,j,k)mid 1 ≤ i < j < k ≤ d, and a basis of K[Xd,Yd]δ as a vector space. As a corollary we have shown that the defining relations R ∩ S form the reduced Gröbner basis of the ideal which they generate with respect to a specific admissible order. This is an analogue of the result of Makar-Limanov and the author in their proof of the Nowicki conjecture in 2009. The algebras K[Xd,Yd]δ, d < 4, are also described. [ABSTRACT FROM AUTHOR]

Let \mathcal {A} denote a central hyperplane arrangement of rank n in affine space \mathbb {K}^n over a field \mathbb {K} of characteristic zero and let l_1,\ldots, l_m\in R≔\mathbb {K}[x_1,\ldots,x_n] denote the linear forms defining the corresponding hyperplanes, along with the corresponding defining polynomial f≔l_1\cdots l_m\in R. The focus of the paper is on the ideal J_f\subset R generated by the partial derivatives of f. We conjecture that J_f is a minimal reduction of the ideal \mathbb {I}\subset R generated by the (m-1)-fold products of distinct forms among l_1,\ldots, l_m. We prove this conjecture for an almost generic \mathcal {A} (i.e., any n-1 among the defining linear forms are linearly independent). In this case we obtain a stronger version of a result by Dimca and Papadima, and we confirm the conjecture unconditionally for n=3. We also conjecture that J_f is an ideal of linear type (i.e., the respective symmetric and Rees algebras coincide). We prove this conjecture for n=3. In the sequel we explain the tight relationship between the two ideals J_f, \mathbb {I}\subset R; in particular, we show that in the generic case (J_f)^{\text {sat}}=\mathbb I. As a consequence, we can provide a simpler proof of a conjectured result of Yuzvinsky, proved by Rose and Terao, on the vanishing of the depth of R/J_f. [ABSTRACT FROM AUTHOR]

The equivariant Kazhdan-Lusztig polynomial of a matroid was introduced by Gedeon, Proudfoot, and Young. Gedeon conjectured an explicit formula for the equivariant Kazhdan-Lusztig polynomials of thagomizer matroids with an action of symmetric groups. In this paper, we discover a new formula for these polynomials which is related to the equivariant Kazhdan- Lusztig polynomials of uniform matroids. Based on our new formula, we confirm Gedeon's conjecture by the Pieri rule. [ABSTRACT FROM AUTHOR]

We give a self-contained proof of the strongest version of Mason's conjecture, namely that for any matroid the sequence of the number of independent sets of given sizes is ultra log-concave. To do this, we introduce a class of polynomials, called completely log-concave polynomials, whose bivariate restrictions have ultra log-concave coefficients. At the heart of our proof we show that for any matroid, the homogenization of the generating polynomial of its independent sets is completely log-concave. [ABSTRACT FROM AUTHOR]

From Carlitz's identity, we deduce two new q-supercongruences modulo the square of a cyclotomic polynomial, which were originally conjectured by Guo. These results establish new q-analogues of a supercongruence of Sun. [ABSTRACT FROM AUTHOR]

We prove the dual Smale's mean value conjecture for polynomials of degree six: if f is a polynomial of degree six with f(0) = 0 and f'(0) = 1, then there is a point ζ such that f'(ζ) = 0 and ∣f(ζ)/ζ∣ ≥ 1/6. [ABSTRACT FROM AUTHOR]

Sendov's conjecture says that if all zeros of a complex polynomial P lie in the closed unit disk and a denotes one of them, then the closed disk of center a and radius 1 contains a critical point of P (i.e. a zero of its derivative P'). The main result of this paper is to prove that, for each a, there exists an integer N such that the disk |ζ -a| ≤ 1 contains a critical point of P when the degree of P is larger than N. We obtain this by studying the geometry of the zeros and critical points of a polynomial which would eventually contradict Sendov's conjecture. [ABSTRACT FROM AUTHOR]

We give positive answer to two conjectures posed by M. E. H Ismail in his monograph [ Classical and quantum orthogonal polynomials in one variable , Cambridge University Press, Cambridge, 2005]. These results generalize the classical problems of Sonine and Hahn. [ABSTRACT FROM AUTHOR]

Suppose f \in K[x] is a polynomial. The absolute Galois group of K acts on the preimage tree \mathrm {T} of 0 under f. The resulting homomorphism \phi _f\colon \operatorname {Gal}_K \to \operatorname {Aut} \mathrm {T} is called the arboreal Galois representation. Odoni conjectured that for all Hilbertian fields K there exists a polynomial f for which \phi _f is surjective. We show that this conjecture is false. [ABSTRACT FROM AUTHOR]

A recent nice result due to I. Pak and G. Panova is the strict unimodality of the q-binomial coefficients ... . Since their proof used representation theory and Kronecker coefficients, the authors also asked for an argument that would employ Zeilberger's KOH theorem. In this note, we give such a proof. Then, as a further application of our method, we also provide a short proof of their conjecture that the difference between consecutive coefficients of ... can get arbitrarily large, when we assume that b is fixed and a is large enough. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Hansen-Mullen conjecture with the help of some formal series similar to the Artin-Hasse exponential series over $p$-adic number fields and the estimates of character sums over Galois rings. Given $n$ we prove, for large enough $q$, the Hansen-Mullen conjecture that there exists a primitive polynomial $f(x)=x^{n}-a_{1}x^{n-1}+\cdots +(-1)^{n}a_{n}$ over $ F_{q}$ of degree $n$ with the $m$-th ($0

We give an explicit formula for the degree of the Grothendieck polynomial of a Grassmannian permutation and a closely related formula for the Castelnuovo-Mumford regularity of the Schubert determinantal ideal of a Grassmannian permutation. We then provide a counterexample to a conjecture of Kummini-Lakshmibai-Sastry-Seshadri on a formula for regularities of standard open patches of particular Grassmannian Schubert varieties and show that our work gives rise to an alternate explicit formula in these cases. We end with a new conjecture on the regularities of standard open patches of arbitrary Grassmannian Schubert varieties. [ABSTRACT FROM AUTHOR]

We show that the real parts of the roots of the Alexander polynomial for any 2-bridge knot are greater than -1. This is a specific case of Hoste's conjecture. [ABSTRACT FROM AUTHOR]

Let I be an ideal of a polynomial algebra over a field generated by r square free monomials of degree d. If r is bigger than (or equal to, if I is not principal) the number of square free monomials of I of degree d + 1, then depthSI = d. Let J ⊊ I, J ≠ 0 be generated by square free monomials of degree = d + 1. If r is bigger than the number of square free monomials of I \ J of degree d + 1 or, more generally, the Stanley depth of I/J is d, then depthSI/J = d. In particular, Stanley's Conjecture holds in these cases. [ABSTRACT FROM AUTHOR]

Consider a representation ρ: L → gl(V) where L is a Lie algebra and V is a finite dimensional vector space. We prove the analog of Amitsur's conjecture on asymptotic behavior for codimensions of polynomial identities of ρ. [ABSTRACT FROM AUTHOR]

For an odd integer n > 0, we introduce the class LPn of Laurent polynomials Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. with all coefficients ck equal to -1 or 1. Such polynomials arise in the study of Barker sequences of even length, i.e., integer sequences having minimal possible autocorrelations. We prove that polynomials P ε LPn have large Mahler measures, namely, M(P) > (n + 1)/2. We conjecture that minimal Mahler measures in the class LPn are attained by the polynomials Rn(z) and Rn(-z), where Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. is a polynomial with all the coefficients ck = 1. We prove that Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. The results of experimental computations on polynomials in the class LPn suggest two conjectures which could shed light on the long-standing Barker problem. [ABSTRACT FROM AUTHOR]