MotivicHomotopy.jl

MotivicHomotopy.MotivicHomotopy — Module

MotivicHomotopy

Computing local and global $\mathbb{A}^1$-Brouwer degrees and studying the symmetric bilinear forms they produce, over $\mathbb{Q}$, finite fields of odd characteristic, $\mathbb{R}$, and $\mathbb{C}$.

The $\mathbb{A}^1$-Brouwer degree of an endomorphism of affine space with isolated zeros is a class in the Grothendieck–Witt ring $\text{GW}(k)$ of symmetric bilinear forms; for pointed rational functions $\mathbb{P}^1_k → \mathbb{P}^1_k$ the degree lives in the unstable group $\text{GW}^u(k)$. The package provides:

Classes: GWClass, GWuClass with accessors (gw_matrix, gw_scalar, gw_algebra, gw_base_field, stable_part) and operations (gw_direct_sum, gw_tensor_product, divisorial_sum).
Form constructors: diagonal_form, hyperbolic_form, pfister_form and unstable variants.
Simplification: diagonalize_via_congruence, diagonal_class, diagonal_entries, sum_decomposition, sum_decomposition_string, anisotropic_part.
Invariants: form_rank, form_signature, integral_discriminant, hilbert_symbol_padic, hilbert_symbol_real, hasse_witt_invariant, anisotropic_dimension, gw_witt_index, is_isotropic_form, is_anisotropic_form, is_isomorphic_form.
Degrees: global_A1_degree, local_A1_degree, global_unstable_A1_degree, local_unstable_A1_degree.
Étale algebras: multiplication_matrix, algebra_trace, algebra_norm, transfer_gw.

Which arithmetic runs is determined by the base field. Over $\mathbb{Q}$ and finite fields of odd characteristic everything is computed exactly. Over $\mathbb{C}$ a form is determined by its rank, so the stable degrees are the identity form of the algebra dimension (a discrete invariant from the exact computation); only the unstable degrees, which carry a $k^×$-scalar, use numerical (HomotopyContinuation) computation over $\mathbb{C}$. Over $\mathbb{R}$ the degree functions direct the user to compute over $\mathbb{Q}$ and base-change the result.

This package is a Julia port of the Macaulay2 package A1BrouwerDegrees (v2.0); v1.1 is published in the Journal of Software for Algebra and Geometry (14, 2024). It was ported by Claude Fable 5 (Anthropic), with the Macaulay2 package as the authoritative specification for mathematical behavior.

source

Installation

using Pkg
Pkg.add("MotivicHomotopy")

Quick start

using Oscar, MotivicHomotopy

# The A¹-degree of z ↦ z² : rank 2 (complex degree), signature 0 (real degree)
S, (x,) = polynomial_ring(QQ, ["x"])
beta = global_A1_degree([x^2 + 1])
form_rank(beta), form_signature(beta)        # (2, 0)

# Local degrees sum to the global degree
S, (x, y) = polynomial_ring(QQ, ["x", "y"])
f = [x^3 - x^2 - y, y]
d1 = local_A1_degree(f, ideal(S, [x - 1, y]))
d2 = local_A1_degree(f, ideal(S, [x, y]))
is_isomorphic_form(global_A1_degree(f), gw_direct_sum(d1, d2))   # true

# Witt decomposition of a form
sum_decomposition_string(GWClass(QQ[1 2 3; 2 4 5; 3 5 6]))       # "H + <1>"

The unstable degrees compute numerically over $\mathbb{C}$ when given HomotopyContinuation expressions (the stable degrees are exact everywhere — over $\mathbb{C}$ they are just the identity form of the algebra dimension):

import HomotopyContinuation; HomotopyContinuation.@var x
global_unstable_A1_degree((x-1)*(x-2)*(x-3), (x-1)*(x-4))

Over $\mathbb{R}$ the degree functions ask you to compute over $\mathbb{Q}$ and base-change the result. Forms over $\mathbb{R}$ / $\mathbb{C}$ are represented by Matrix{Float64} / Matrix{ComplexF64} Gram matrices; pass Float64 / ComplexF64 where a base field is expected.

See the API reference for every exported function, grouped by topic.

Conventions

== is literal Gram-matrix equality; mathematical equality in $\text{GW}(k)$ is is_isomorphic_form.
Accessors and operations whose natural names would collide with Oscar exports carry a gw_ prefix (gw_matrix, not gram_matrix; gw_witt_index, not witt_index).
Degree Gram matrices are expressed in a standard-monomial basis; an equivalent form in another basis is the same class, so compare with is_isomorphic_form rather than entrywise.

Version history

MotivicHomotopy.jl began as a Julia port of the Macaulay2 package A1BrouwerDegrees. Its initial release consolidates the functionality of that package's versions 1.1 and 2.0 into a single starting point; subsequent releases listed below build on it.

v0.1.0 — based on versions 1.1 and 2.0 of the Macaulay2 package A1BrouwerDegrees. Implements the computation of local and global 𝔸¹-Brouwer degrees and of Grothendieck–Witt classes and their invariants; the computation of unstable local and global 𝔸¹-Brouwer degrees and manipulation of the unstable Grothendieck–Witt group; and the extension of Grothendieck–Witt class manipulations over fields to finite étale algebras over fields, including transfers along finite étale extensions. Developed by S. Atherton, N. Borisov, T. Brazelton, S. Dutta, F. Espino, T. Hagedorn, Z. Han, J. Lopez Garcia, J. Louwsma, Y. Luo, G. Ong, R. Sagayaraj, and A. Tawfeek.