feat(Algebra/Ring/Subring/Polynomial): existence of polynomials that can represent elements of a subring

[FMLJohn]

---

In this pull request, I have mainly proved the two results:

1. Suppose R is a commutative ring, x is an element of R, and A is a subring of R. If y is an element of the subring of R generated by A and x, then there exists a univariate polynomial p with coefficents in A such that p(x)=y.
2. Suppose R is a commutative ring, S is a subset of R, and A is a subring of R. If r is an element of the subring of R generated by A and S, then there exists a multivariate polynomial p with coefficents in A such that p(S)=r.

[github-actions]

PR summary bdc43c9104

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Algebra.Ring.Subring.Polynomial (new file)	1161

---

Declarations diff (regex)

+ exists_mvPolynomial_of_le_range_of_subset_range_of_mem_closure
+ exists_mvPolynomial_of_mem_closure
+ exists_mvPolynomial_of_mem_closure₁
+ exists_polynomial_of_le_range_of_mem_closure
+ exists_polynomial_of_mem_closure
+ exists_polynomial_of_mem_closure₁

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

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

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

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit bdc43c9).

• +6 new declarations
• −0 removed declarations

+Subring.exists_mvPolynomial_of_le_range_of_subset_range_of_mem_closure
+Subring.exists_mvPolynomial_of_mem_closure
+Subring.exists_mvPolynomial_of_mem_closure₁
+Subring.exists_polynomial_of_le_range_of_mem_closure
+Subring.exists_polynomial_of_mem_closure
+Subring.exists_polynomial_of_mem_closure₁

---

No changes to strong technical debt.

Increase in weak tech debt: (relative, absolute) = (1.00, 0.00)

Current number	Change	Type (weak)
4925	1	exposed public sections

Current commit bdc43c9104
Reference commit ba6dea3d06

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/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).

[plp127]

Suggested change

	@[expose] public section
	public section

There are no definition, so no need to @[expose].

[plp127]

I think normally, instead of having {h : A →+* R}, you would add a [Algebra A R] assumption and remove h. Then you can also remove B and have hy : y ∈ Algebra.adjoin A {x}. And actually, after you make these changes, it becomes identical to the existing lemma Algebra.adjoin_mem_exists_aeval.