Effective Nullstellensatz and Generalized B ezout identities Andr - PowerPoint PPT Presentation

Effective Nullstellensatz and Generalized B´ ezout identities Andr´ e GALLIGO (U.C.A., INRIA, LJAD, France) JNCF , CIRM February 2019. A NDR ´ E G ALLIGO , N ICE Effective Nullstellensatz and Generalized B´ ezout identities

Abstract Among recent results on effective Hilbert’s Nullstellensatz: Z. Jelonek (Inventiones mathematicae, 2005) C. d’Andrea, T. Krick and M. Sombra (A. S. ENS, 2013) “[DKS:13]”. I will present our curent work with Z. Jelonek, for finding effective versions of sharp elimination processes. A NDR ´ E G ALLIGO , N ICE Effective Nullstellensatz and Generalized B´ ezout identities

Hilbert Nullstellensatz f 1 , . . . , f s ∈ C [ x 1 , . . . , x n ] do not share any root in C n if and only if there exist g 1 , . . . , g s ∈ C [ x 1 , . . . , x n ] such that: 1 = g 1 f 1 + . . . + g s f s . Assuming deg ( f i ) ≤ d . If the degrees of the f i g i , is bounded by D , one finds the g i by solving a linear system of size about sD n . The coefficients of the g i belong to the field of coefficients of the f i , (e.g. Q ). A NDR ´ E G ALLIGO , N ICE Effective Nullstellensatz and Generalized B´ ezout identities

Brief History: Upper bound D for the degrees Hermann, 1923: D = 2 ( 2 d ) 2 n − 1 . Brownawell, 1987: D = n 2 d n , in characteristic 0. Caniglia-Galligo-Heintz, 1988: D = d n ( n + 3 ) / 2 . Kollar, 1988: D = max ( d , 3 ) n . Fitchas-Giusti-Smietanski, 1995: D = d cn , for a constant c. (Using Straight-Line Programs). Sabia-Solerno, Sombra, 1995-97: Improvements for d = 2. Jelonek, 2005: D = d n , for s ≤ n . C. d’Andrea, T. Krick and M. Sombra, 2013: Parametric and arithmetic versions. A NDR ´ E G ALLIGO , N ICE Effective Nullstellensatz and Generalized B´ ezout identities

Elimination and B´ ezout identities Let K be an algebraically closed field. When V ( f 1 , . . . , f s ) is of dimension 0 in K n , Z. Jelonek established in 2005, an elimination theorem. We generalize this result as follows. Assume V ( f 1 , . . . , f s ) has dimension q in K n ; deg ( f 1 ) ≥ . . . ≥ deg ( f s ) . There exist g 1 , . . . , g s ∈ C [ x ] and a non-zero polynomial φ ( x n − q , . . . , x n ) , such that: φ = g 1 f 1 + . . . + g s f s ; deg ( g i f i ) ≤ [ deg ( f 1 ) . . . deg ( f n − q − 1 )] deg ( f n ) . We first prove it in generic coordinates, then we use a deformation argument. A NDR ´ E G ALLIGO , N ICE Effective Nullstellensatz and Generalized B´ ezout identities

Perron’s theorem Jelonek type approaches rely on generalizations of Perron’s theorem. Here, we will use one proved in [DKS:13]. Let k be an arbitrary field and consider the groups of variables t = { t 1 , . . . , t p } and x = { x 1 , . . . , x n } . Generalized Perron Theorem: Let Q 1 , . . . , Q n + 1 ∈ k [ t , x ] \ k [ t ] . d = ( d 1 , ..., d n + 1 ) , h = ( h 1 , . . . , h n + 1 ) . Then there exists α a y a ∈ k [ t ][ y 1 , . . . , y n + 1 ] \ { 0 } � E = a ∈ N n + 1 satisfying E ( Q 1 , . . . , Q n + 1 ) = 0 and such that, for all a ∈ supp ( E ) , we have 1 ) < d , a > ≤ ( � n + 1 i = 1 d j ) . 2 ) deg ( α a )+ < h , a > ≤ ( � n + 1 i = 1 d j )( � n + 1 h l d l ) . l = 1 A NDR ´ E G ALLIGO , N ICE Effective Nullstellensatz and Generalized B´ ezout identities

Main Construction I = ( f 1 , . . . , f s ) ⊂ K [ x 1 , . . . , x n ] is an ideal, of dimension q < n . Take F n − q = f s and F i = � s j = i α ij f j for i = 1 , ..., n − q − 1 , where α ij are sufficiently general. Take J = ( F 1 , ..., F n − q ) , deg F n − q = d s , deg F i = d i for i ≤ n − q − 1, dimV ( J ) = q . Φ : K n × K ∋ ( x , z ) → ( F 1 ( x ) z , . . . , F n − q ( x ) z , x ) ∈ K n − q × K n is a (non-closed) embedding outside the set V ( J ) × K . Γ = cl (Φ( K n × K )) is an affine sub-variety of dimension n + 1 of K 2 n − q . Let π : Γ → K n + 1 be a generic projection and define Ψ := π ◦ Φ . In the generic coordinates X , we get Ψ( X , z ) = ( zF 1 + ℓ 0 ( x ) , zF 2 + X 1 . . . , zF n − q + X n − q − 1 , X n − q , . . . , X n ) . A NDR ´ E G ALLIGO , N ICE Effective Nullstellensatz and Generalized B´ ezout identities

Continued By this genericity, the image of the projection π ′ : V ( J ) ∋ X �→ ( X n − q , . . . , X n ) ∈ K q + 1 is an hypersurface S , let φ ′ ( X n − q , . . . , X n ) = 0 describe S . Ψ = (Ψ 1 , . . . , Ψ n − q , X n − q , . . . , X n ) : K n × K → K n + 1 is finite outside the set V ( J ) × K . Hence, the set of non-properness of Ψ is contained in S = { T = ( T 1 , . . . , T n − q , X n − q , . . . , X n ) ∈ K n + 1 : φ ′ ( X ) = 0 } . Now, we apply to Ψ , Perron’s theorem with L = K ( z ) . There exists a non-zero polynomial W ( T 1 , . . . , T n + 1 ) ∈ L [ T 1 , . . . , T n + 1 ] such that W (Ψ 1 , . . . , Ψ n + 1 ) = 0 with the expected degree inequalities. A NDR ´ E G ALLIGO , N ICE Effective Nullstellensatz and Generalized B´ ezout identities

End of proof There is a non-zero minimal poynomial ˜ W ∈ K [ T 1 , . . . , T n + 1 , Y ] such that (a) ˜ W (Ψ 1 ( x , z ) , . . . , Ψ n + 1 ( x , z ) , z ) = 0 , 2 , . . . , T d n − q (b) deg T ˜ W ( T d 1 1 , T d 2 n − q , T n − q + 1 , . . . , T n + 1 , Y ) ≤ � n − q − 1 d s d j , j = 1 The Y − leading coefficient b 0 ( T ) of ˜ W satisfies { T : b 0 ( T ) = 0 } ⊂ S , hence b 0 ( T ) depends only on coordinates T n − q + i + 1 = X n − q + i , for 0 ≤ i ≤ q . We now develop (a) in z and get E ( X , z ) = 0. The z − leading coefficient B ( X ) in E , is obtained either from b 0 ( X n − q , ..., X n ) or from terms corresponding to products, containing at least one of T i , i < n , hence containing at least one of F i . The B´ ezout identity follows from the fact that this coefficient B ( X ) vanishes identically. � A NDR ´ E G ALLIGO , N ICE Effective Nullstellensatz and Generalized B´ ezout identities

Getting rid of the coordinates change We first establish a parametric version: We replace the field K by the algebraic closure of the fraction field of k [ t ] , where k is an infinite field, following [DKS:13]. Then, we use the following generic change of coordinates and its inverse. n n � � X i = x i + t a i , j x j ; x i = X i + t b i , j ( t ) X j . j = i + 1 j = i + 1 Set ¯ F j ( X , t ) = F j ( x ) . Notice that t divises ¯ F j ( X , t ) − F j ( X ) . After simpliflications, we have, n − q G j ( X , t ) ¯ � b 0 ( X n − q , ..., X n , t ) = F j ( X , t ) . j = 1 A NDR ´ E G ALLIGO , N ICE Effective Nullstellensatz and Generalized B´ ezout identities

Continuation We cannot exclude the possibility of a remaining factor t p in the left hand, side with p > 0. So we need to perform several reduction steps. Let b 0 ( X , t )) = t p ( φ ( x ) + t φ 1 ( x , t )) . Setting t = 0, we obtain a non trivial relation 0 = � s j = 1 G j ( x , 0 ) F j ( x ) . Apply a change of coordinates to this relation to get 0 = � s j = 1 ¯ H j ( X , t ) ¯ F j ( X , t ) . The x − degree of G j ( x , 0 ) is bounded by the X − degree of G j ( X , t ) , and is equal to the X − degree of ¯ H j ( X , t ) . Now, � n − q j = 1 ( G j ( X , t ) − ¯ H j ( X , t )) ¯ F j ( X , t ) vanishes for t = 0, hence admits a factor t . We simplify the two sides of the previous equality by t , so t p − 1 ( φ ( x ) + t φ 1 ( x , t )) = � s j = 1 ( G j ( X , t ) − ¯ H j ( X , t )) ¯ F j ( X , t ) . � A NDR ´ E G ALLIGO , N ICE Effective Nullstellensatz and Generalized B´ ezout identities