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Mathlib.RingTheory.MvPolynomial.Basic

Multivariate polynomials over commutative rings #

This file contains basic facts about multivariate polynomials over commutative rings, for example that the monomials form a basis.

Main definitions #

Main statements #

TODO #

Generalise to noncommutative (semi)rings

instance MvPolynomial.instCharP (σ : Type u) (R : Type v) [CommSemiring R] (p : ) [CharP R p] :
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theorem MvPolynomial.mapRange_eq_map (σ : Type u) {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] (p : MvPolynomial σ R) (f : R →+* S) :
def MvPolynomial.restrictSupport {σ : Type u} (R : Type v) [CommSemiring R] (s : Set (σ →₀ )) :

The submodule of polynomials that are sum of monomials in the set s.

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    restrictSupport R s has a canonical R-basis indexed by s.

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      The submodule of polynomials of total degree less than or equal to m.

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        The submodule of polynomials such that the degree with respect to each individual variable is less than or equal to m.

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          theorem MvPolynomial.mem_restrictTotalDegree (σ : Type u) {R : Type v} [CommSemiring R] (m : ) (p : MvPolynomial σ R) :
          p MvPolynomial.restrictTotalDegree σ R m p.totalDegree m
          theorem MvPolynomial.mem_restrictDegree (σ : Type u) {R : Type v} [CommSemiring R] (p : MvPolynomial σ R) (n : ) :
          p MvPolynomial.restrictDegree σ R n sp.support, ∀ (i : σ), s i n
          theorem MvPolynomial.mem_restrictDegree_iff_sup (σ : Type u) {R : Type v} [CommSemiring R] [DecidableEq σ] (p : MvPolynomial σ R) (n : ) :
          p MvPolynomial.restrictDegree σ R n ∀ (i : σ), Multiset.count i p.degrees n

          The monomials form a basis on MvPolynomial σ R.

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            @[simp]
            instance MvPolynomial.instFree (σ : Type u) (R : Type v) [CommSemiring R] :

            The R-module MvPolynomial σ R is free.

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            theorem MvPolynomial.linearIndependent_X (σ : Type u) (R : Type v) [CommSemiring R] :
            LinearIndependent R MvPolynomial.X
            noncomputable def Polynomial.basisMonomials (R : Type v) [CommSemiring R] :

            The monomials form a basis on R[X].

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