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January 20, 2026Open Access

Discriminant Loci, Galois Symmetry, and Cm Specializations in a Cubic Family Arising From an Arc–difference Elimination Problem

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PEParker Emmerson

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Overview

Symbolic elimination addresses cubic equations and their geometric properties, indicating implications for complex multiplication.

Key Points

  • The aim is to explore a symbolic elimination problem's outputs through algebraic and geometric analysis of cubic equations.
  • Derived a mixed-derivative identity and a degree-12 univariate elimination polynomial.
  • Parametrized a locus based on the discriminant of a cubic polynomial.
  • Attained a short Weierstrass model for a genus-1 curve based on the cubic equation.
  • Embedded the cubic polynomial into a two-parameter deformation to analyze complex multiplication.
  • Identified a discriminant that yields a rational conic for certain conditions.
  • Showed that the Weierstrass model does not have complex multiplication for positive rational parameters.
  • Determined a finite set of rational specializations with complex multiplication invariant.

Cite This Study

Parker Emmerson (2026) studied this question.

synapsesocial.com/papers/696f1ac19e64f732b51eef88https://doi.org/10.5281/zenodo.18275581
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