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[Merged by Bors] - feat(ring_theory/polynomial): Vieta's formula in terms of polynomial.roots
#14908
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urkud
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…m_esymm for multiset (#15008) This is a proof of Vieta's formula for `multiset`: ```lean lemma multiset.prod_X_add_C_eq_sum_esymm (s : multiset R) : (s.map (λ r, polynomial.X + polynomial.C r)).prod = ∑ j in finset.range (s.card + 1), (polynomial.C (s.esymm j) * polynomial.X ^ (s.card - j)) ``` where ```lean def multiset.esymm (s : multiset R) (n : ℕ) : R := ((s.powerset_len n).map multiset.prod).sum ``` From this, we deduce the proof of the formula for `mv_polynomial`: ```lean lemma prod_C_add_X_eq_sum_esymm : (∏ i : σ, (polynomial.X + polynomial.C (X i)) : polynomial (mv_polynomial σ R) )= ∑ j in range (card σ + 1), (polynomial.C (esymm σ R j) * polynomial.X ^ (card σ - j)) ``` with ```lean def mv_polynomial.esymm (n : ℕ) : mv_polynomial σ R := ∑ t in powerset_len n univ, ∏ i in t, X i ``` It is better to go in this direction since there does not seem to be a natural way to prove the `multiset` version from the `mv_polynomial` version due to the difficulty to enumerate the elements of a `multiset` with a `fintype`, see this [Zulip thread](https://leanprover.zulipchat.com/#narrow/stream/113489-new-members/topic/.E2.9C.94.20fintype.20enumerating.20a.20multiset) and the discussion from #14908. We prove, as suggested in [#14908](#14908 (comment)) , that `mv_polynomial.esymm` is obtained by specialising `multiset.esymm`. However, we do not refactor the results of `ring_theory/polynomial/symmetric` in terms of `multiset.esymm`. From flt-regular
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….roots` (#14908) Specialize `multiset.prod_X_sub_C_coeff` to the root multiset of a split polynomial over an integral domain to derive the familiar Vieta's formula; update *undergrad.yaml*. Make various stylistic improvements to *polynomial/vieta.lean*: most notably, `open polynomial` to be able to omit the prefix in `polynomial.X` and `polynomial.C`. Instead, write `mv_polynomial.X` with the prefix because it's less frequent. Prove miscellaneous lemmas `list.prod_map_neg`, `multiset.prod_map_neg`, `list.map_nth_le` and `multiset.length_to_list`, which are remnants of a previous approach to prove `polynomial.vieta` superseded by #15008. See below/#14908 for the original motivation for introducing them.
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Specialize
multiset.prod_X_sub_C_coeffto the root multiset of a split polynomial over an integral domain to derive the familiar Vieta's formula; update undergrad.yaml.Make various stylistic improvements to polynomial/vieta.lean: most notably,
open polynomialto be able to omit the prefix inpolynomial.Xandpolynomial.C. Instead, writemv_polynomial.Xwith the prefix because it's less frequent.Prove miscellaneous lemmas
list.prod_map_neg,multiset.prod_map_neg,list.map_nth_leandmultiset.length_to_list, which are remnants of a previous approach to provepolynomial.vietasuperseded by #15008. See below/#14908 for the original motivation for introducing them.Below is the PR description before #15008 was merged:
In order to derive the multiset version of
prod_C_add_X_coefffrom the existing fintype-indexed version, we need to index (with repetition) the elements of a multisetsusing a functionffrom a fintypeα, such that(@finset.univ α _).val.map f = s, see Zulip.I originally opt to use
α = fin s.cardbut finally switched to @kmill's multiset.has_coe_to_sort. The new lemmasmap_nth_lein list/range.lean andlength_to_listin multiset/basic.lean were needed originally but no longer now, but I think they're worth having.Add new big_operator lemma
list/multiset.prod_map_negto facilitate transition frommultiset.prod_C_add_X_coefftomultiset.prod_X_sub_C_coeff.