[Merged by Bors] - feat(Algebra/Polynomial): add aevalEquiv and its bivariate version

[Multramate]

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[github-actions]

PR summary 7c0aa26a8b

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

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Declarations diff

+ aevalAevalEquiv
+ aevalAevalEquiv_apply
+ aevalAeval_C
+ aevalAeval_X
+ aevalAeval_Y
+ aevalEquiv
+ aevalEquiv_apply

You can run this locally as follows

## summary with just the declaration names:
./scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh contains some details about this script.

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No changes to technical debt.

You can run this locally as

./scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[Copilot]

Pull request overview

This PR adds two new equivalences that characterize algebra homomorphisms from polynomial algebras: Polynomial.algHomEquiv establishes a bijection between R[X] →ₐ[R] A and A, while Polynomial.Bivariate.algHomEquiv establishes a bijection between R[X][Y] →ₐ[R] A and A × A. These equivalences formalize the universal property of polynomial algebras.

Changes:

• Added Polynomial.algHomEquiv equivalence for univariate polynomials
• Added Polynomial.Bivariate.algHomEquiv equivalence for bivariate polynomials with supporting lemmas

Reviewed changes

Copilot reviewed 2 out of 2 changed files in this pull request and generated 1 comment.

File	Description
Mathlib/Algebra/Polynomial/AlgebraMap.lean	Adds algHomEquiv establishing the bijection (R[X] →ₐ[R] A) ≃ A
Mathlib/Algebra/Polynomial/Bivariate.lean	Adds helper lemmas (aevalAeval_C, aevalAeval_X, aevalAeval_Y) and Bivariate.algHomEquiv establishing the bijection (R[X][Y] →ₐ[R] A) ≃ A × A

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[Copilot]

The docstring has a typo: it should be "A × A` induced" instead of "A × A induced" (missing the closing backtick after "A × A").

Suggested change

	/-- The bijection `(R[X][Y] →ₐ[R] A) ≃ A × A induced by `Polynomial.aevalAeval`. -/
	/-- The bijection `(R[X][Y] →ₐ[R] A) ≃ A × A` induced by `Polynomial.aevalAeval`. -/

[mathlib-merge-conflicts]

This pull request has conflicts, please merge master and resolve them.

[ocfnash]

bors merge

[mathlib-bors]

Build failed (retrying...):

• ci (staging) / Lint style

[Multramate]

@ocfnash I'm happy with my new feature additions but I'm not 100% sure about changing the simp normal form for aeval and aevalAeval etc - could you confirm that you're also happy with them?

[mathlib-bors]

Pull request successfully merged into master.

Build succeeded:

• ci (staging) / Post-CI job
• ci (staging) / Post-Build Step
• ci (staging) / Lint style
• ci (staging) / Build

[ocfnash]

@ocfnash I'm happy with my new feature additions but I'm not 100% sure about changing the simp normal form for aeval and aevalAeval etc - could you confirm that you're also happy with them?

Unfortunately I'm too late reading this. If you have specific concerns, please ping me in a follow-up PR.