feat(Algebra): add multivariate Laurent polynomials

[yuma-mizuno]

---

I used Codex to imitate the MvPolynomial API. Now it has been rewritten by hand to focus on its AddMonoidAlgebra features.
Proving UniqueFactorizationMonoid is my next plan.

[github-actions]

PR summary 28a77d90ec

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Algebra.MvLaurentPolynomial.Basic (new file)	1065

---

Declarations diff

+ X
+ X_eq_monomial
+ X_inj
+ X_injective
+ X_ne_zero
+ X_pow_eq_monomial
+ algebraMap_eq_toMvLaurent
+ algebraMvPolynomial
+ as_sum
+ basisAlgEquiv
+ basisAlgEquiv_apply
+ basisAlgEquiv_monomial
+ basisAlgEquiv_toMvLaurent
+ induction_on'
+ isUnit_monomial
+ monomial
+ monomial_add_single
+ monomial_eq_zero
+ monomial_left_inj
+ monomial_left_injective
+ monomial_mul
+ monomial_pow
+ monomial_single_add
+ monomial_zero
+ mul_eq_sum_monomial
+ one_eq_monomial
+ single_eq_monomial
+ smul_monomial
+ sum_monomial_eq
+ support_monomial
+ support_monomial_subset
+ support_sum_monomial_coeff
+ toMvLaurent
+ toMvLaurent_X
+ toMvLaurent_eq_zero
+ toMvLaurent_inj
+ toMvLaurent_injective
+ toMvLaurent_monomial
+ toMvLaurent_ne_zero

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.

---

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).

[YaelDillies]

Can you please reuse as many AddMonoidAlgebra primitives as possible? In particular, could you please avoid defining: coeff, support. monomial?

My hope is that you can work with mutlvariate polynomials without adding any API at all, but I can see that maybe you want MvLaurentPolynomial.C and MvLaurentPolynomial.X, the latter of which doesn't make sense over AddMonoidAlgebra.

[YaelDillies]

Suggested change

	abbrev MvLaurentPolynomial (σ : Type*) (R : Type*) [CommSemiring R] :=
	abbrev MvLaurentPolynomial (σ R : Type*) [CommSemiring R] :=

[yuma-mizuno]

My hope is that you can work with mutlvariate polynomials without adding any API at all, but I can see that maybe you want MvLaurentPolynomial.C and MvLaurentPolynomial.X, the latter of which doesn't make sense over AddMonoidAlgebra.

Indeed, what I actually want would be X and its related things. Can we develop the theory of Laurent polynomials as a part of AddMonoidAlgebra API by using the definitions like this?

def X {σ R M : Type*} [CommSemiring R] [AddCommGroup M] [Module ℤ M]
    (v : Basis σ ℤ M) (i : σ) :
    AddMonoidAlgebra R M :=
  AddMonoidAlgebra.single (v i) 1

[YaelDillies]

Possibly, yes. In Toric, we need "multivariate polynomials without a canonical basis". @erdOne, @Kiolt, do you think what Yuma is suggesting above would be useful to us?