Uniform Model Completeness for the Real Field with the Weierstrass $\wp$ Function

TL;DR

This paper proves that the expansion of the real field by the Weierstrass  function and related functions is model complete, involving complex analysis, special functions, and auxiliary results, but does not establish effectiveness.

Contribution

It establishes model completeness for the real field expanded by the Weierstrass  function, including derivatives and related functions, using a novel approach.

Findings

Model completeness is achieved for the expansion with  and related functions.

Auxiliary results show model completeness with composed functions.

The proof is non-effective and highlights challenges for effective results.

Abstract

In this work is we prove model completeness for the expansion of the real field by the Weierstrass $\wp$ function as a function of the variable $z$ and the parameter (or period) $\tau$. We need to existentially define the partial derivatives of the $\wp$ function with respect to the variable $z$ and the parameter $\tau$. In order to obtain this result we need to include in the structure function symbols for the unrestricted exponential function and restricted sine function, the Weierstrass $\zeta$ function and the quasimodular form $E_2$. We prove some auxiliary model completeness results with the same functions composed with appropriate change of variables. In the conclusion we make some remarks about the noneffectiveness of our proof and the difficulties to be overcome to obtain an effective model completeness result.