In this paper we introduce a new model for computing polynomials a depth 2 circuit with a symmetric gate at the top and plus gates at the bottom, i.e the circuit computes a symmetric function in linear functions Sd m(`1; `2; :::; `m) (Sd m is the d’th elementary symmetric polynomial in m variables, and the `i’s are linear functions). We refer to this model as the symmetric model. This new model...

A standard way of treating the polynomial eigenvalue problem P (λ)x = 0 is to convert it into an equivalent matrix pencil—a process known as linearization. Two vector spaces of pencils L1(P ) and L2(P ), and their intersection DL(P ), have recently been defined and studied by Mackey, Mackey, Mehl, and Mehrmann. The aim of our work is to gain new insight into these spaces and the extent to which...

Some new formulas related to the well-known symmetric Lucas polynomials are primary focus of this article. Different approaches used for establishing these formulas. A matrix approach is followed in order obtain some fundamental properties. Particularly, recurrence relations and determinant forms determined by suitable Hessenberg matrices. Conjugate generating functions derived examined. Severa...

This paper studies the elementary symmetric polynomials Sk(x) for x ∈ Rn. We show that if |Sk(x)|, |Sk+1(x)| are small for some k > 0 then |S`(x)| is also small for all ` > k. We use this to prove probability tail bounds for the symmetric polynomials when the inputs are only t-wise independent, which may be useful in the context of derandomization. We also provide examples of t-wise independent...

In this paper we establish two symmetric identities on sums of products of Euler polynomials.

A set of n homogeneous polynomials in n variables is a regular sequence if the associated polynomial system has only the obvious solution (0, 0, . . . , 0). Denote by pk(n) the power sum symmetric polynomial in n variables x k 1 +x 2 + · · ·+xk n . The interpretation of the q-analogue of the binomial coefficient as Hilbert function leads us to discover that n consecutive power sums in n variabl...

Noncommutative symmetric functions have many properties analogous to those of classical (commutative) symmetric functions. For instance, ribbon Schur functions (analogs of the classical Schur basis) expand positively in noncommutative monomial basis. More of the classical properties extend to noncommutative setting as I will demonstrate introducing a new family of noncommutative symmetric funct...

Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved lower bounds for the rank of a symmetric tensor (i.e., a homogeneous polynomial) obtained by considering the singularities of the hypersurface defined by the polynomial. We obtain normal forms for po...

Recently Lapointe et. al. [3] have expressed Jack Polynomials as determinants in monomial symmetric functions mλ. We express these polynomials as determinants in elementary symmetric functions eλ, showing a fundamental symmetry between these two expansions. Moreover, both expansions are obtained indifferently by applying the Calogero-Sutherland operator in physics or quasi Laplace Beltrami oper...

This behavior has been seen in some notable cases. Kirillov [3] shows that elementary symmetric polynomials in noncommuting variables commute (and, in some cases, all Schur functions) when elementary symmetric polynomials of degree at most three commute when restricted to at most three of the variables. Generalizing this, Blasiak and Fomin [1] give a wider theory for rules of three of generatin...