From a Polynomial Riemann Hypothesis to Alternating Sign Matrices

This paper begins with a brief discussion of a class of polynomial Riemann hypotheses, which leads to the consideration of sequences of orthogonal polynomials and 3-term recursions. The discussion further leads to higher order polynomial recursions, including 4-term recursions where orthogonality is lost. Nevertheless, we show that classical results on the nature of zeros of real orthogonal polynomials (i. e., that the zeros of $p_n$ are real and those of $p_{n+1}$ interleave those of $p_n$) may be extended to polynomial sequences satisfying certain 4-term recursions. We identify specific polynomial sequences satisfying higher order recursions that should also satisfy this classical result. As with the 3-term recursions, the 4-term recursions give rise naturally to a linear functional. In the case of 3-term recursions the zeros fall nicely into place when it is known that the functional is positive, but in the case of our 4-term recursions, we show that the functional can be positive even when there are non-real zeros among some of the polynomials. It is interesting, however, that for our 4-term recursions positivity is guaranteed when a certain real parameter $C$ satisfies $C\ge 3$, and this is exactly the condition of our result that guarantees the zeros have the aforementioned interleaving property. We conjecture the condition $C\ge 3$ is also necessary. Next we used a classical determinant criterion to find exactly when the associated linear functional is positive, and we found that the Hankel determinants $\Delta_n$ formed from the sequence of moments of the functional when $C = 3$ give rise to the initial values of the integer sequence $1, 3, 26, 646, 45885, \cdots,$ of Alternating Sign Matrices (ASMs) with vertical symmetry. This spurred an intense interest in these moments, and we give 9 diverse characterizations of this sequence of moments. We then specify these Hankel determinants as Macdonald-type integrals. We also provide an an infinite class of integer sequences, each sequence of which gives the Hankel determinants $\Delta_n$ of the moments. Finally we show that certain $n$-tuples of non-intersecting lattice paths are evaluated by a related class of special Hankel determinants. This class includes the $\Delta_n$. At the same time, ASMs with vertical symmetry can readily be identified with certain $n$-tuples of osculating paths. These two lattice path models appear as a natural bridge from the ASMs with vertical symmetry to Hankel determinants.