Algebro-geometric aspects of Heine-Stieltjes theory

作者：Shapiro Boris^*

来源：Journal of the London Mathematical Society-Second Series, 2011, 83: 36-56.

DOI：10.1112/jlms/jdq061

The goal of this paper was to develop a Heine-Stieltjes theory for univariate linear differential operators of higher order. For a given linear ordinary differential operator partial derivative(z) = Sigma(k)(i=1) Q(i)(z)(d(i)/dz(i)) with polynomial coefficients, set r = max(i=1, ... , k)(deg Q(i)(z) - i). If partial derivative(z) satisfies conditions (i) r >= 0 and (ii) deg Q(k)(z) = k + r, then we call it a non-degenerate higher Lame operator. Following the classical approach of Heine [Handbuch der Kugelfunctionen 1 (G. Reimer, Berlin, 1878) 472-479] and Stieltjes ['Sur certains polynomes qui verifient une equation differentielle lineaire du second ordre et sur la theorie des fonctions de Lame, Acta Math. 6 (1885) 321-326 (French)], we study the multiparameter spectral problem of finding all polynomials V(z) of degree at most r such that for a given positive integer n, the equation
partial derivative(z)S(z) + V(z)S(z) = 0
has a polynomial solution S(z) of degree n. We show that under some mild non-degeneracy assumptions, there exist exactly ((n+r)(n)) such polynomials V(n,i)(z) whose corresponding eigenpolynomials S(n,i)(z) are of degree n. We generalize a number of well-known results in this area and discuss occurring degeneracies.

• 出版日期2011-2