Tangential polynomials and elliptic solitons
Let \(\Gamma\) and X be two integral projective curves over \({\mathbb{C}}\), \(p\in \Gamma\) and \(q\in X\) two smooth points and \(\pi:\quad (\Gamma,p)\to (X,q)\) a finite pointed morphism \((\pi (p)=q)\). We call \(\pi\) a ``tangential cover'' if and only if the canonical images of (\(\Gamma\),p) and (X,q) in the generalized jacobian of \(\Gamma\) are tangent at the origin. If X has arithmetic genus equal to 1 (i.e.: X is an elliptic curve or has one node or one cusp) then the solutions of the KP equation associated by the Krichever correspondence to the ``source'' of \(\pi\), (\(\Gamma\),p), are elliptic, trigonometric or rational solitons. We start our article characterizing a tangential cover \(\pi:\quad (\Gamma,p)\to (X,q)\) by the existence of a rational function on \(\Gamma\) (called here ``tangential'') having specific polar developments over q (the latter ``tangency criterion'' was first proved by Krichever in case X is an elliptic curve, \(\Gamma\) is smooth and \(\pi\) is etale over q). We then consider the characteristic polynomial P(T) of the ``tangential'' function. We prove that the coefficients of P(T) are rational functions on X, holomorphic on X-\(\{\) \(q\}\) such that for some local coordinate, z, of X at q, all the coefficients of \(zP(T-z^{-1})\) are holomorphic at q. Conversely, any polynomial P(T) satisfying the above conditions (is called ``tangential'' and) corresponds canonically (and quite effectively) to a tangential cover, equipped with a tangential function whose characteristic polynomial coincides with P(T). Having made clear the usefulness of the tangential polynomials we develop two different constructions of these polynomials. The first one by induction on the degree and the other one as image of \({\mathbb{C}}[T]\) by the natural action of a series operator \(1+\sum_{m\geq 1}\alpha_ m(z)\Delta^ m \), \(\{\alpha_ m(z); m\geq 1\}\) being a set of particular rational functions on X and \(\Delta =\partial /\partial T\) the differentiation with respect to the variable T.
scientific article; zbMATH DE number 4144137
scientific article