[Merged by Bors] - feat(algebraic_geometry/EllipticCurve/weierstrass): define Weierstrass polynomials and prove basic facts #17631
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…polynomials (#17664) + To show a monic polynomial `p` over a commutative semiring without zero divisors is irreducible, it suffices to show that `p ≠ 1` and that there doesn't exist a factorization into the product of two monic polynomials of positive degrees: `monic.irreducible_iff_nat_degree`. + If such a factorization exists, one of the polynomial must have degree less than or equal to `p.nat_degree / 2`: `monic.irreducible_iff_nat_degree'`. + To show a quadratic monic polynomial `p` is reducible, it suffices to it doesn't factor as `(X+c)(X+c')`, i.e. there don't exist `c` and `c'` that add up to the 1st coefficient and multiply up to the 0th coefficient: `not_irreducible_iff_exists_add_mul_eq_coeff`. + Define the ring homomorphism `constant_coeff` in *coeff.lean* which is analogous to [mv_polynomial.constant_coeff](https://leanprover-community.github.io/mathlib_docs/data/mv_polynomial/basic.html#mv_polynomial.constant_coeff); use it to golf and generalize `is_unit_C` and `eq_one_of_is_unit_of_monic` in *monic.lean* and various lemmas in *ring_division.lean*. + Add `monic.is_unit_iff`. + Golf some lemmas in *erase_lead.lean*. Co-authored-by: Thomas Browning <thomas.l.browning@gmail.com> Co-authored-by: Anne Baanen <t.baanen@vu.nl> Inspired by #17631
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Co-authored-by: Junyan Xu <junyanxumath@gmail.com>
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…s polynomials and prove basic facts (#17631) Define the Weierstrass polynomial and its partial derivatives, as well as properties of their evaluations (the Weierstrass equation and nonsingularity). Prove that a Weierstrass curve is nonsingular at every point if its discriminant is non-zero, and its coordinate ring is an integral domain (because the associated polynomial is irreducible). Fix minor issues (rename the variable `C` with the variable `W` to avoid a clash with `polynomial.C`, and generalise `two_torsion_polynomial_disc_ne_zero`). Co-authored-by: Junyan Xu <junyanxumath@gmail.com>
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Define the Weierstrass polynomial and its partial derivatives, as well as properties of their evaluations (the Weierstrass equation and nonsingularity). Prove that a Weierstrass curve is nonsingular at every point if its discriminant is non-zero, and its coordinate ring is an integral domain (because the associated polynomial is irreducible). Fix minor issues (rename the variable
Cwith the variableWto avoid a clash withpolynomial.C, and generalisetwo_torsion_polynomial_disc_ne_zero).Co-authored-by: Junyan Xu junyanxumath@gmail.com