Complete Symmetric Polynomials Research Articles

On an integral representation of the normalized trace of the k-th symmetric tensor power of matrices and some applications

Let A be an matrix and let be its k-th symmetric tensor power. We express the normalized trace of as an integral of the k-th powers of the numerical values of A over the unit sphere of with respect to the rotation-invariant probability measure. Equivalently, this expression in turn can be interpreted as an integral representation for the (normalized) complete symmetric polynomials over . As applications, we present a new proof for the MacMahon Master Theorem in enumerative combinatorics. Then, our next application deals with a generalization of the work of Cuttler et al. in [Cuttler A, Greene C, Skandera M. Inequalities for symmetric means. Eur J Comb. 2011;32(6):745–761] concerning the monotonicity of products of complete symmetric polynomials. Finally, we give a solution to an open problem that was raised by Rovena and Temereanca in [Roventa I, Temereanca LE. A note on the positivity of the even degree complete homogeneous symmetric polynomials. Mediterr J Math. 2019;16(1):1–16].

Waring and cactus ranks and strong Lefschetz property for annihilators of symmetric forms

In this note we show that the complete symmetric polynomials are dual generators of compressed artinian Gorenstein algebras satisfying the Strong Lefschetz Property. This is the first example of an explicit dual form with these properties. For complete symmetric forms of any degree in any number of variables, we provide an upper bound for the Waring rank by establishing an explicit power sum decomposition. Moreover, we determine the Waring rank, the cactus rank, the resolution and the Strong Lefschetz Property for any Gorenstein algebra defined by a symmetric cubic form. In particular, we show that the difference between the Waring rank and the cactus rank of a symmetric cubic form can be made arbitrarily large by increasing the number of variables. We provide upper bounds for the Waring rank of generic symmetric forms of degrees four and five.

Complete symmetric polynomials over finite fields have many rational zeros

Counting zeros of polynomials over finite fields is one of the most important topics in arithmetic algebraic geometry. In this paper, we consider the problem for complete symmetric polynomials. The homogeneous complete symmetric polynomial of degree $m$ in the $k$-variables $\\{x_1,x_2,\\ldots,x_k\\}$is defined as <disp-formula><tex-math><![CDATA[$$h_m(x_1,x_2,\\ldots,x_k):=\\sum_{1\\leq i_1\\leq i_2\\leq \\cdots \\leq i_m\\leq k}x_{i_1}x_{i_2}\\cdots x_{i_m}.$$]]></tex-math></disp-formula> A complete symmetric polynomial of degree $m$ over $\\f{q}$ in the $k$-variables $\\{x_1,x_2,\\ldots,x_k\\}$ is defined as <disp-formula><tex-math><![CDATA[$$h(x_1,\\ldots, x_k):=\\sum_{e=0}^m a_eh_e(x_1,x_2,\\ldots,x_k),$$]]></tex-math></disp-formula> where $a_e\\in~\\f{q}$ and $a_m\\not=0$. We are interested in counting the number of zeros and the number of zeros with pairwise distinct coordinates of a complete symmetric polynomial, respectively. Let <disp-formula><tex-math><![CDATA[$$N_q(h):= \\#\\{(a_1,\\ldots, a_k)\\in \\f{q}^k\\mid h(a_1,\\ldots, a_k)=0\\}$$]]></tex-math></disp-formula> denote the number of $\\f{q}$-rational points on theaffine hypersurface defined by $h(x_1,\\ldots,~x_k)=0$. Let <disp-formula><tex-math><![CDATA[$$N_q^*(h):=\\# \\{(a_1,\\ldots, a_k)\\in \\f{q}^k\\mid h(a_1,\\ldots, a_k)=0 \\mbox{and}\\, a_i\\not= a_j, \\forall\\, i\\not=j\\}$$]]></tex-math></disp-formula> denote the number of $\\f{q}$-rational points on theaffine hypersurface defined by $h(x_1,\\ldots,~x_k)=0$with the additional condition that the coordinates are distinct. In the paper Zhang and Wan (2020), the authors showed the lower bound $N_q(h)\\geq~6q^{k-3}$ if $q$ is odd, $k\\geq~3$ and $1\\leq~m\\leq~k-3$ and conjectured $N_q(h)\\geq~24q^{k-4}$ if $q$ is even, $k\\geq~4$ and $1\\leq~m\\leq~k-4.$ The key ingredient in the proof of the lower bound is to prove $N_q^*(h(x_1,x_2,x_3))\\geq~6$ for odd $q$, $k=3$ and $1\\leq~m\\leq~q-3$ which does not hold for even $q$ in general. In this paper, we deal with the even characteristic case. The main new results are the followings (suppose $\\f{q}$ is a finite field with characteristic $2$). (1) Let $h(x_1,x_2,~x_3):=\\sum_{e=0}^m~a_eh_e(x_1,x_2,x_3)\\in~\\f{q}[x_1,x_2,x_3]$ be a complete symmetric polynomials of degree $m$ with $1\\leq~m\\leq~q-3$. If $a_{e_0}\\neq~0$ for some odd $e_0$, then $N_q^*(h)\\geq~6.$ (2) Let $h(x_1,\\ldots,~x_k)$ be a complete symmetric polynomials in $k\\geq~3$ variables over $\\f{q}$ of any degree $m$ with $1\\leq~m\\leq~q-3$. If $m$ is odd, then $N_q(h)\\geq~6q^{k-3}.$ (3) Let $h(x_1,\\ldots,~x_k)$be a complete symmetric polynomial in $k\\geq~4$ variables over $\\f{q}$ ofdegree $m$ with $1\\leq~m\\leq~q-3$. Then $N_q(h)\\geq~6(q-1)q^{k-4}.$ (4) As a consequence, Conjecture 1.7 in the paper [Zhang~J,~Wan~D.~Rational~points~on~complete~symmetric~hypersurfaces~over~finite~fields.~Discrete~Math,~2020,~343:~112072] is true. That is, for any complete symmetric polynomial $h(x_1,\\ldots,~x_k)$ in $k\\geq~4$ variables over $\\f{q}$ ofdegree $m$ with $1\\leq~m\\leq~q-4$, we have$N_q(h)\\geq~24q^{k-4}.$

Rational points on complete symmetric hypersurfaces over finite fields

For any affine hypersurface defined by a complete symmetric polynomial in k≥3 variables of degree m over the finite field Fq of q elements, a special case of our theorem says that this hypersurface has at least 6qk−3 rational points over Fq if 1≤m≤q−3 and q is odd. A key ingredient in our proof is Segre’s classical theorem on ovals in finite projective planes.

Remarks on two symmetric polynomials and some matrices

By means of complete symmetric polynomials this paper gives a new proof for the Vandermonde determinant formula. Another alternative proof for this formula is obtained via the collocation matrices. It also gives a generalized relationship between the Vandermonde, the Pascal and the Stirling matrices. A new approach to obtain the explicit inverse of the Vandermonde matrix is investigated. Closed form expressions for some Vandermonde related determinants are obtained.

Degree $k$ linear recursions $\mod(p)$ and number fields

Linear recursions of degree $k$ are determined by evaluating the sequence of Generalized Fibonacci Polynomials, $\{F_{k,n}(t_1,...,t_k)\}$ (isobaric reflects of the complete symmetric polynomials) at the integer vectors $(t_1,...,t_k)$. If $F_{k,n}(t_1,...,t_k) = f_n$, then $$f_n - \sum_{j=1}^k t_j f_{n-j} = 0,$$ and $\{f_n\}$ is a linear recursion of degree $k$. On the one hand, the periodic properties of such sequences modulo a prime $p$ are discussed, and are shown to be related to the prime structure of certain algebraic number fields; for example, the arithmetic properties of the period are shown to characterize ramification of primes in an extension field. On the other hand, the structure of the semilocal rings associated with the number field is shown to be completely determined by Schur-hook polynomials. Keywords: Symmetric polynomials, Schur polynomials, linear recursions, number fields.

Noncommutative Schur polynomials and the crystal limit of the -vertex model

Starting from the Verma module of we consider the evaluation module for affine and discuss its crystal limit (q → 0). There exists an associated integrable statistical mechanics model on a square lattice defined in terms of vertex configurations. Its transfer matrix is the generating function for noncommutative complete symmetric polynomials in the generators of the affine plactic algebra, an extension of the finite plactic algebra first discussed by Lascoux and Schützenberger. The corresponding noncommutative elementary symmetric polynomials were recently shown to be generated by the transfer matrix of the so-called phase model discussed by Bogoliubov, Izergin and Kitanine. Here we establish that both generating functions satisfy Baxter's TQ-equation in the crystal limit by tying them to special solutions of the Yang–Baxter equation. The TQ-equation amounts to the well-known Jacobi–Trudi formula leading naturally to the definition of noncommutative Schur polynomials. The latter can be employed to define a ring which has applications in conformal field theory and enumerative geometry: it is isomorphic to the fusion ring of the Wess–Zumino–Novikov–Witten model whose structure constants are the dimensions of spaces of generalized θ-functions over the Riemann sphere with three punctures.

Notes on particular symmetric polynomials with applications

This paper presents some applications using several properties of three important symmetric polynomials: elementary symmetric polynomials, complete symmetric polynomials and the power sum symmetric polynomials. The applications includes a simple proof of El-Mikkawy conjecture in [M.E.A. El-Mikkawy, Appl. Math. Comput. 146 (2003) 759–769] and a very easy proof of the Newton–Girard formula. In addition, a generalization of Stirling numbers is obtained.

Generalized Leibniz functional matrices and factorizations of some well-known matrices

In this paper, we introduce the generalized Leibniz functional matrices and study some algebraic properties of such matrices. To demonstrate applications of these properties, we derive several novel factorization forms of some well-known matrices, such as the complete symmetric polynomial matrix and the elementary symmetric polynomial matrix. In addition, by applying factorizations of the generalized Leibniz functional matrices, we redevelop the known results of factorizations of Stirling matrices of the first and second kind and the generalized Pascal matrix.

Symmetric multivariate Chebyshev polynomials

We introduce a general method for the symmetrization of univariate polynomials and use it to construct symmetric polynomials in r + 1 variables that generalize the classical Chebyshev polynomials of the first kind. We show that, on the set [−1, 1] r+1 , such polynomials provide the best uniform approximations of the complete symmetric polynomials h λ ( x 0, x 1, …, x r ) by means of symmetric polynomials of total degree less than n, where λ is a partition of n.