We consider the problem of deciding existence real roots real-valued exponential polynomials with algebraic coefficients. Such functions arise as solutions linear differential equations focus on two problems: Zero Problem , which asks whether an polynomial has a root, and Infinite Zeros such function infinitely many roots. Our main result is that for order at most 8 decidable, subject to Schanu...

Suppose $$\{P_{n}^{(\alpha , \beta )}(x)\} _{n=0}^\infty $$ is a sequence of Jacobi polynomials with \alpha >-1.$$ We discuss special cases question raised by Alan Sokal at OPSFA in 2019, namely, whether the zeros P_{n}^{(\alpha ,\beta )}(x)$$ and P_{n+k}^{(\alpha + t, s are interlacing if $$s,t >0$$ k \in {\mathbb {N}}.$$ consider two this for consecutive degree prove that P_{n+1}^{(\alpha 1 )...

We evaluate the number of complex monic polynomials, arbitrary degree N, zeros which are equal to their coefficients. In following, we call polynomials with this property peculiar polynomials. further show that problem determining N simplifies when any coefficients is either 0 or 1. proceed estimate numbers having one coefficient zero, one, neither.

The partition function of the Potts model can be expressed in terms of famous combinatorial graph polynomials for restricted Hamiltonians. We investigate the Hamiltonian with variable edge interaction and external magnetic field, for which the partition function is the V-polynomial. Because the partition function is a polynomial, we can use motivic methods from algebraic geometry to characteriz...