Let $e$ and $q$ be fixed co-prime integers satisfying $1\lt e\lt q$. Let $\mathscr {C}$ be a certain family of deformations of the curve $y^e=x^q$. That family is called the $(e,q)$-curve and is one of the types of curves called plane telescopic curves. Let $\varDelta$ be the discriminant of $\mathscr {C}$. Following pioneering work by Buchstaber and Leykin (BL), we determine the canonical basis $\{ L_j \}$ of the space of derivations tangent to the variety $\varDelta =0$ and describe their specific properties. Such a set $\{ L_j \}$ gives rise to a system of linear partial differential equations (heat equations) satisfied by the function $\sigma (u)$ associated with $\mathscr {C}$, and eventually gives its explicit power series expansion. This is a natural generalisation of Weierstrass’ result on his sigma function. We attempt to give an accessible description of various aspects of the BL theory. Especially, the text contains detailed proofs for several useful formulae and known facts since we know of no works which include their proofs.

Thermochronometer data offer a powerful tool for quantifying a wide range of geologic processes, such as the deformation and erosion of mountain ranges, topographic evolution, and hydrocarbon maturation. With increasing interest to quantify a wider range of complicated geologic processes, more sophisticated techniques are needed. This paper is concerned with an inverse problem method for interpreting the thermochronometer data quantitatively. Two novel models are proposed to simulate the crustal thermal fields and paleo mountain topography as a function of tectonic and surface processes. One is a heat transport model that describes the change of temperature of rocks; while the other is surface process model which explains the change of mountain topography. New computational algorithms are presented for solving the inverse problem of the coupled system of these two models. The model successfully provides a new tool for reconstructing the kinematic and the topographic history of mountains.