An explicit a priori bound for the condition number associated to each of the following problems is given: general linear equation solving, least squares, non-symmetric eigenvalue problems, solving univariate polynomials, and solving systems of multivariate polynomials. It is assumed that the input has integer coefficients and is not on the degeneracy locus of the respective problem (i.e., the ...

We define a family of polynomials of the form ∑ f(σ)x1,σ(1) · · ·xn,σ(n) in terms of the Kazhdan-Lusztig basis {C′ w(1) |w ∈ Sn} for the symmetric group algebra C[Sn]. Using this family, we obtain nonnegativity properties of polynomials of the form ∑ cI,I′∆I,I′(x)∆I,I′(x). In particular, we show that the application of certain of these polynomials to Jacobi-Trudi matrices yields symmetric funct...

Elementary symmetric polynomials can be thought of as derivative polynomials of En(x) = ∏ i=1,...,n xi. Their associated hyperbolicity cones give a natural sequence of relaxations for R+. We establish a recursive structure for these cones, namely, that the coordinate projections of these cones are themselves hyperbolicity cones associated with elementary symmetric polynomials. As a consequence ...

G. E. Murphy showed in 1983 that the centre of every symmetric group algebra has an integral basis consisting of a specific set of monomial symmetric polynomials in the Jucys–Murphy elements. While we have shown in earlier work that the centre of the group algebra of S3 has exactly three additional such bases, we show in this paper that the centre of the group algebra of S4 has infinitely many ...

We study the problem of representing symmetric Boolean functions as symmetric polynomials over Zm. We show an equivalence between such representations and simultaneous communication protocols. Computing a function f on 0 1 inputs with a polynomial of degree d modulo pq is equivalent to a two player simultaneous protocol for computing f where one player is given the first dlogp de digits of the ...