Positivity Conditions for Quartic Polynomials

Splitting of Quartic Polynomials

For integers r, s, t, u define the recursion A(n + 4) = rA(n + 3) sA(n + 2) + tA(n + 1) uA(n) where the initial conditions are set up in such a way that A(n) = a" + ß" + y" + S" where a, ß, y, S are the roots of the associated polynomial f(x) = x* rxi + sx2 tx + u. In this paper a detailed deterministic procedure using the A(n) for finding how f(x) splits modulo a prime integerp is given. This ...

Newton’s Method for Symmetric Quartic Polynomials

We investigate the parameter plane of the Newton’s method applied to the family of quartic polynomials pa,b(z) = z 4 +az + bz +az+ 1, where a and b are real parameters. We divide the parameter plane (a, b) ∈ R into twelve open and connected regions where p, p′ and p′′ have simple roots. In each of these regions we focus on the study of the Newton’s operator acting on the Riemann sphere.

Positivity Of Trigonometri Polynomials

A positivity conjecture for Jack polynomials

We present a positivity conjecture for the coefficients of the development of Jack polynomials in terms of power sums. This extends Stanley’s ex-conjecture about normalized characters of the symmetric group. We prove this conjecture for partitions having a rectangular shape.

Positivity of Turán determinants for orthogonal polynomials

The orthogonal polynomials pn satisfy Turán’s inequality if p 2 n(x)− pn−1(x)pn+1(x) ≥ 0 for n ≥ 1 and for all x in the interval of orthogonality. We give general criteria for orthogonal polynomials to satisfy Turán’s inequality. This yields the known results for classical orthogonal polynomials as well as new results, for example, for the q–ultraspherical polynomials.