General explicit formula for the Cauchy kernel on arbitrary plane curves

Construct an explicit uniform formula for the Cauchy kernel _{D−p_o}(p,q) on an arbitrary plane algebraic curve E(x,y)=0, where _{D−p_o}(p,q) is the unique third–kind differential in p with zeros at the non–special divisor D and simple poles at q and p_o of residues ±1, expressed rationally in the curve’s coefficients and the coordinates of the involved points, and applicable without assuming coprimeness of the leading and subleading coefficients with respect to y or x.

Background

The paper develops an algebraic reconstruction of rational Lax matrices from spectral data using a Cauchy kernel, avoiding transcendental objects like theta functions. A key step is providing an explicit, computable expression for the Cauchy kernel subordinated to a divisor.

While the author gives a rational expression for the Cauchy kernel under the assumption that the leading and subleading coefficients in y (or x) have no common root, they note the absence of a general formula covering all plane curves without such assumptions. Establishing such a formula would remove these restrictions and fully generalize the algebraic reconstruction approach.