(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
From mathcomp
From mathcomp
From mathcomp
(******************************************************************************)
(* This file provides additional primitives and theory for bivariate          *)
(* polynomials (polynomials of two variables), represented as polynomials     *)
(* with (univariate) polynomial coefficients :                                *)
(*            'Y == the (generic) second variable (:= 'X%:P).                 *)
(*          p^:P == the bivariate polynomial p['X], for p univariate.         *)
(*               := map_poly polyC p (this notation is defined in poly.v).    *)
(*      u.[x, y] == the bivariate polynomial u evaluated at 'X = x, 'Y = y.   *)
(*               := u.[x%:P].[y].                                             *)
(*       sizeY u == the size of u in 'Y (1 + the 'Y-degree of u, if u != 0).  *)
(*               := \max_(i < size u) size u`_i.                              *)
(*      swapXY u == the bivariate polynomial u['Y, 'X], for u bivariate.      *)
(*    poly_XaY p == the bivariate polynomial p['X + 'Y], for p univariate.    *)
(*               := p^:P \Po ('X + 'Y).                                       *)
(*    poly_XmY p == the bivariate polynomial p['X * 'Y], for p univariate.    *)
(*               := P^:P \Po ('X * 'Y).                                       *)
(* sub_annihilant p q == for univariate p, q != 0, a nonzero polynomial whose *)
(*                  roots include all the differences of roots of p and q, in *)
(*                  all field extensions (:= resultant (poly_XaY p) q^:P).    *)
(* div_annihilant p q == for polynomials p != 0, q with q.[0] != 0, a nonzero *)
(*                  polynomial whose roots include all the quotients of roots *)
(*                  of p by roots of q, in all field extensions               *)
(*                  (:= resultant (poly_XmY p) q^:P).                         *)
(* The latter two "annhilants" provide uniform witnesses for an alternative   *)
(* proof of the closure of the algebraicOver predicate (see mxpoly.v). The    *)
(* fact that the annhilant does not depend on the particular choice of roots  *)
(* of p and q is crucial for the proof of the Primitive Element Theorem (file *)
(* separable.v).                                                              *)
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Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Notation "p ^ f" := (map_poly f p) : ring_scope.
Local Notation eval := horner_eval.
Notation "'Y" := 'X%:P : ring_scope.
Notation "p ^:P" := (p ^ polyC) (at level 2, format "p ^:P") : ring_scope.
Notation "p .[ x , y ]" := (p.[x%:P].[y])
(at level 2, left associativity, format "p .[ x ,  y ]") : ring_scope.
Section PolyXY_Ring.
Variable R : ringType.
Implicit Types (u : {poly {poly R}}) (p q : {poly R}) (x : R).
Fact swapXY_key : unit. Proof. by []. Qed.
Definition swapXY_def u : {poly {poly R}} := (u ^ map_poly polyC).['Y].
Definition swapXY := locked_with swapXY_key swapXY_def.
Canonical swapXY_unlockable := [unlockable fun swapXY].
Definition sizeY u : nat := \max_(i < size u) (size u`_i).
Definition poly_XaY p : {poly {poly R}} := p^:P \Po ('X + 'Y).
Definition poly_XmY p : {poly {poly R}} := p^:P \Po ('X * 'Y).
Definition sub_annihilant p q := resultant (poly_XaY p) q^:P.
Definition div_annihilant p q := resultant (poly_XmY p) q^:P.

Lemma poly_XaY0 : poly_XaY 0 = 0.
Section PolyXY_ComRing.