Kronecker's and Newton's approaches to solving: A first comparison

TL;DR

This work analyzes two paradigms for solving multivariate diophantine polynomial systems over number fields: Kronecker's symbolic geometric solution and Newton's numerical approximate-zero method. It develops a unified framework under common hypotheses, extending Smale's theory to diophantine contexts, defining $\gamma$-theorem quantities $\gamma_\nu(F,\zeta)$ and the universal bound $\widetilde{\gamma}(F,\zeta)$, and examining height and complexity via $LLL$-reduction and Hensel lifting to connect symbolic and numeric viewpoints. It demonstrates that Kronecker's method can be made practical with straight-line programs and non-archimedean lifting, achieving polynomial-time results on structured instances, and presents a hybrid algorithm that computes splitting fields and Lagrange resolvents by combining both approaches. Overall, the paper bridges symbolic algebraic geometry and numerical diophantine approximation, informing effective number theory and computational algebraic geometry.

Abstract

In this extended abstract we deal with the relations between the numerical/diophantine approximation and the symbolic/algebraic geometry approachs to solving of multivariate diophentine polynomial systems, obtaining several consecuences ranging from diophantine approximation to effective number theory.

Theorems & Definitions (22)

• Definition 1
• Theorem 2: $\gamma-$Theorem
• Theorem 3: Eckardt & Young
• Theorem 4: Lower Bounds
• Proposition 5
• Corollary 6
• Corollary 7
• Example 1: Using $\log\gamma$ as in Theorem \ref{['theorem-lower-bounds']}
• Corollary 8
• Corollary 9