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Motivic configurations on the line

John Igieobo, Stephen McKean, Steven Sanchez, Dae'Shawn Taylor, Kirsten Wickelgren

TL;DR

This work defines a family of configuration-dependent operations $ ext{sum}_D$ on the unstable $ ext{P}^1$-loop space and analyzes their images under the unstable degree map $ ext{deg}^u$ to the unstable Grothendieck–Witt group $ ext{GW}^u(k)$. A local-to-global framework is developed: unstable local degrees at zeros are computed via local Newton matrices, yielding an explicit global formula for $ ext{deg}^u( ext{sum}_D(f_1, ots,f_n))$ that factors through a discriminant-like term $igotimes_{i<j}(r_i-r_j)^{2m_i m_j}$. The paper introduces the divisorial $D$-sum $igoplus_D$ and proves that its image under $ ext{deg}^u$ is governed by the duplicant $ rak{D}(f)$, refining naive localizations and aligning with Morel’s anabelian perspective on $ ext{P}^1$. The key technical contributions combine Bézoutian formalisms, local Newton matrices, and a two-step anabelian argument via $ ext{pi}_1^{ ext{A}^1}( ext{P}^1)$ to prove a robust local-to-global principle for unstable degrees. The results illuminate how motivic homotopy-theoretic constructions interact with quadratic form invariants and discriminants, revealing discriminant-sensitive behavior absent in the classical topological setting.

Abstract

For each configuration of rational points on the affine line, we define an operation on the group of unstable A1 motivic homotopy classes of endomorphisms of the projective line. We also derive an algebraic formula for the image of such an operation under Cazanave and Morel's unstable degree map, which is valued in an extension of the Grothendieck--Witt group. In contrast to the topological setting, these operations depend on the choice of configuration of points via a discriminant. We prove this by first showing a local-to-global formula for the global unstable degree as a modified sum of local terms. We then use an anabelian argument to generalize from the case of local degrees of a global rational function to the case of an arbitrary collection of endomorphisms of the projective line.

Motivic configurations on the line

TL;DR

This work defines a family of configuration-dependent operations on the unstable -loop space and analyzes their images under the unstable degree map to the unstable Grothendieck–Witt group . A local-to-global framework is developed: unstable local degrees at zeros are computed via local Newton matrices, yielding an explicit global formula for that factors through a discriminant-like term . The paper introduces the divisorial -sum and proves that its image under is governed by the duplicant , refining naive localizations and aligning with Morel’s anabelian perspective on . The key technical contributions combine Bézoutian formalisms, local Newton matrices, and a two-step anabelian argument via to prove a robust local-to-global principle for unstable degrees. The results illuminate how motivic homotopy-theoretic constructions interact with quadratic form invariants and discriminants, revealing discriminant-sensitive behavior absent in the classical topological setting.

Abstract

For each configuration of rational points on the affine line, we define an operation on the group of unstable A1 motivic homotopy classes of endomorphisms of the projective line. We also derive an algebraic formula for the image of such an operation under Cazanave and Morel's unstable degree map, which is valued in an extension of the Grothendieck--Witt group. In contrast to the topological setting, these operations depend on the choice of configuration of points via a discriminant. We prove this by first showing a local-to-global formula for the global unstable degree as a modified sum of local terms. We then use an anabelian argument to generalize from the case of local degrees of a global rational function to the case of an arbitrary collection of endomorphisms of the projective line.

Paper Structure

This paper contains 14 sections, 31 theorems, 170 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Outline
  3. Terminology and notation
  4. Unstable Grothendieck--Witt groups
  5. Bézoutians
  6. Unstable local degree
  7. Algebraic formula for the unstable local degree
  8. Local-to-global principle, homotopically
  9. Aside on duplicants
  10. Local-to-global principle, algebraically
  11. Insufficiency of the naïve local-to-global principle
  12. Divisorial sums of local terms
  13. Proof of Theorem \ref{['thm:D-sum-all-f_i']}
  14. Code

Key Result

Theorem 1.1

Let $D=\{r_1,\ldots,r_n\}\subset\mathbb{A}^1_k(k)$. For any unstable pointed $\mathbb{A}^1$-homotopy classes of maps $f_1,\ldots,f_n\in[\mathbb{P}^1,\mathbb{P}^1]$, we have where $(\beta_i, d_i)=\deg^u(f_i)$ and $m_i = \operatorname{rank} \deg^u(f_i)$ for each $i$.

Figures (1)

  • Figure 1: Pinching $S^1$

Theorems & Definitions (89)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • ...and 79 more