Introduction to D -module Theory. Algorithms for Computing Bernstein-Sato Polynomials Jorge Mart´ ın-Morales Centro Universitario de la Defensa de Zaragoza Academia General Militar Differential Algebra and Related Topics October 27-30, 2010, Beijing, China J. Mart´ ın-Morales (jorge@unizar.es) Algorithms for Computing Bernstein-Sato Polynomials
Joint work with . . . Preprint available at arXiv:1003.3478v1 [math.AG]. Viktor Levandovskyy (RWTH Aachen University, Germany) Daniel Andres J. Mart´ ın-Morales (jorge@unizar.es) Algorithms for Computing Bernstein-Sato Polynomials
Motivation to Singularity Theory J. Mart´ ın-Morales (jorge@unizar.es) Algorithms for Computing Bernstein-Sato Polynomials
Motivation Let f ∈ C [ x ] = C [ x 1 , . . . , x n ] be a polynomial. p ∈ C n is said to be singular if p ∈ V ( f , ∂ f ∂ x 1 , . . . , ∂ f ∂ x n ). ∂ x ( p )( x − a ) + ∂ f ∂ f ∂ y ( p )( y − b ) = 0 p = ( a, b ) C 2 p f ( p ) = 0 X = f − 1 (0) To study singular points � invariants. Two hypersurfaces X = V ( f ) , Y = V ( g ) ⊆ C n are called algebraically equivalent if there exists an algebraic isomorphism ϕ : X → Y . J. Mart´ ın-Morales (jorge@unizar.es) Algorithms for Computing Bernstein-Sato Polynomials
First part of the talk Basic notations and definitions History of the problem . . . Well-known properties. Algorithms for computing b f ( s ) J. Mart´ ın-Morales (jorge@unizar.es) Algorithms for Computing Bernstein-Sato Polynomials
Basic notations C the field of the complex numbers. C [ s ] the ring of polynomials in one variable over C . C [ x ] = C [ x 1 , . . . , x n ] the ring of polynomials in n variables. D n = C [ x 1 , . . . , x n ] � ∂ 1 , . . . , ∂ n � the ring of C -linear differential operators, i.e. the n -th Weyl algebra: ∂ i x i = x i ∂ i + 1 D n [ s ] = D n ⊗ C C [ s ]. J. Mart´ ın-Morales (jorge@unizar.es) Algorithms for Computing Bernstein-Sato Polynomials
The Weyl algebra �� �� ∂ ∂ W n = C φ x 1 , . . . , φ x n , ∂ x 1 , . . . , ⊂ End C ( C [ x ]) ∂ x 1 − → C [ x 1 , . . . , x n ] C [ x 1 , . . . , x n ] φ x i : f �− → x i f ∂ ∂ f : f �− → ∂ x i ∂ x i C { x 1 , . . . , x n , ∂ 1 , . . . , ∂ n } � � D n = { x i , x j − x j x i , ∂ i ∂ j − ∂ j ∂ i , ∂ i x j − x j ∂ i − δ ij } J. Mart´ ın-Morales (jorge@unizar.es) Algorithms for Computing Bernstein-Sato Polynomials
The Weyl algebra �� �� ∂ ∂ W n = C φ x 1 , . . . , φ x n , ∂ x 1 , . . . , ⊂ End C ( C [ x ]) ∂ x 1 − → C [ x 1 , . . . , x n ] C [ x 1 , . . . , x n ] φ x i : f �− → x i f ∂ ∂ f : f �− → ∂ x i ∂ x i C { x 1 , . . . , x n , ∂ 1 , . . . , ∂ n } � � D n = { x i , x j − x j x i , ∂ i ∂ j − ∂ j ∂ i , ∂ i x j − x j ∂ i − δ ij } Proposition ∂ The natural map x i �→ φ x i , ∂ i �→ is a C -algebra isomorphism ∂ x i between W n and D n . J. Mart´ ın-Morales (jorge@unizar.es) Algorithms for Computing Bernstein-Sato Polynomials
The non-commutative relations come from Leibniz rule. ∂ i x i = x i ∂ i + 1 The set of monomials { x α ∂ β | α, β ∈ N n } forms a basis as C -vector space. � a αβ x α ∂ β ( a αβ ∈ C ) P = α,β To define a D n -module, it is enough to give the action over the generators and then check that the relations are preserved. For any monomial order there exists a Gr¨ obner basis. The Weyl algebra is simple, i.e. there are no two-sided ideals. J. Mart´ ın-Morales (jorge@unizar.es) Algorithms for Computing Bernstein-Sato Polynomials
The D n [ s ] -module C [ x , s , 1 f ] · T (Bernstein, 1972) Let f ∈ C [ x ] be a non-zero polynomial. By C [ x , s , 1 f ] we denote the ring of rational functions of the form g ( x , s ) f k where g ( x , s ) ∈ C [ x , s ] = C [ x 1 , . . . , x n , s ]. We denote by C [ x , s , 1 f ] · T the free C [ x , s , 1 f ]-module of rank one generated by the symbol T . G ( x , s ) · T J. Mart´ ın-Morales (jorge@unizar.es) Algorithms for Computing Bernstein-Sato Polynomials
The D n [ s ] -module C [ x , s , 1 f ] · T (Bernstein, 1972) C [ x , s , 1 f ] · T has a natural structure of left D n [ s ]-module. x i • ( G ( x , s ) · T ) = x i G ( x , s ) · T � ∂ G � + G ( x , s ) s ∂ f 1 ∂ i • ( G ( x , s ) · T ) · T = ∂ x i ∂ x i f s • ( G ( x , s ) · T ) = sG ( x , s ) · T J. Mart´ ın-Morales (jorge@unizar.es) Algorithms for Computing Bernstein-Sato Polynomials
The previous expression defines an action. ( x i x j − x j x i ) • ( G ( x , s ) · T ) = 0 · T ( ∂ i ∂ j − ∂ j ∂ i ) • ( G ( x , s ) · T ) = 0 · T ( ∂ i x j − x j ∂ i ) • ( G ( x , s ) · T ) = 0 · T ( i � = j ) ( ∂ i x i − x i ∂ i − 1) • ( G ( x , s ) · T ) = 0 · T J. Mart´ ın-Morales (jorge@unizar.es) Algorithms for Computing Bernstein-Sato Polynomials
→ T = f s Where does this action come from? − J. Mart´ ın-Morales (jorge@unizar.es) Algorithms for Computing Bernstein-Sato Polynomials
→ T = f s Where does this action come from? − � ∂ G � + G ( x , s ) s ∂ f 1 ∂ i • ( G ( x , s ) · T ) = · T ∂ x i ∂ x i f = ⇐ � ∂ G � + G ( x , s ) s ∂ f 1 ∂ i • ( G ( x , s ) · f s ) = · f s ∂ x i ∂ x i f J. Mart´ ın-Morales (jorge@unizar.es) Algorithms for Computing Bernstein-Sato Polynomials
Classical notation f ] · f s := C [ x , s , 1 C [ x , s , 1 f ] · T f s := 1 · T f s + k := f k · T ( k ∈ Z ) 0 := 0 · T ∂ i • f s = s ∂ f ∂ i • T = s ∂ f 1 f s − 1 = ⇒ f · T ∂ x i ∂ x i J. Mart´ ın-Morales (jorge@unizar.es) Algorithms for Computing Bernstein-Sato Polynomials
The global b -function Theorem (Bernstein, 1972) For every polynomial f ∈ C [ x ] there exists a non-zero polynomial b ( s ) ∈ C [ s ] and a differential operator P ( s ) ∈ D n [ s ] such that � � x , s , 1 P ( s ) • f s +1 = b ( s ) · f s · f s . ∈ C f Definition (Bernstein & Sato, 1972) The set of all possible polynomials b ( s ) satisfying the previos equation is an ideal of C [ s ]. The monic generator of this ideal is denoted by b f ( s ) and called the global Bernstein-Sato polynomial or global b -function. J. Mart´ ın-Morales (jorge@unizar.es) Algorithms for Computing Bernstein-Sato Polynomials
Examples · · · x m k 1 Normal crossing divisor f = x m 1 k . 1 c · ∂ m 1 · · · ∂ m k P ( s ) = 1 k m 1 � � m k � � � � s + i 1 s + i k · · · b f ( s ) = m 1 m k i 1 =1 i k =1 2 The classical cusp f = x 2 + y 3 . 1 x ∂ y + 1 y + 1 4 ∂ x s + 3 12 y ∂ 2 27 ∂ 3 8 ∂ 2 P ( s ) = x � �� �� � s + 5 s + 7 b f ( s ) = s + 1 6 6 J. Mart´ ın-Morales (jorge@unizar.es) Algorithms for Computing Bernstein-Sato Polynomials
The local b -function Now assume that f ∈ O = C { x 1 , . . . , x n } is a convergent power series. D n is the ring of differential operators with coefficients in O . Theorem (Kashiwara & Bj¨ ork, 1976) For every f ∈ O there exists a non-zero polynomial b ( s ) ∈ C [ s ] and a differential operator P ( s ) ∈ D n [ s ] such that � � s , 1 P ( s ) • f s +1 = b ( s ) · f s · f s . ∈ O f Definition The monic polynomial in C [ s ] of lowest degree which satisfies the previous equation is denoted by b f , 0 ( s ) and called the local Bernstein-Sato polynomial of f at the origin or local b -function. J. Mart´ ın-Morales (jorge@unizar.es) Algorithms for Computing Bernstein-Sato Polynomials
Some well-known properties of the b -function 1 The b -function is always a multiple of ( s + 1). The equality holds if and only f is smooth. 2 The (resp. local) Bernstein-Sato polynomial is an (resp. analytic) algebraic invariant of the singularity V = { f = 0 } . 3 The set { e 2 π i α | b f , 0 ( α ) = 0 } coincides with the eigenvalues of the monodromy of the Milnor fibration. (Malgrange, 1975 and 1983). J. Mart´ ın-Morales (jorge@unizar.es) Algorithms for Computing Bernstein-Sato Polynomials
Some well-known properties of the b -function 1 Every root of b f ( s ) is negative and rational. (Kashiwara, 1976). 2 The roots of b f ( s ) belong to the real interval ( − n , 0). (Varchenko, 1980; Saito, 1994). 3 b f ( s ) = lcm p ∈ C n { b f , p ( s ) } = lcm p ∈ Sing ( f ) { b f , p ( s ) } (Brian¸ con-Maisonobe and Mebkhout-Narv´ aez, 1991). J. Mart´ ın-Morales (jorge@unizar.es) Algorithms for Computing Bernstein-Sato Polynomials
Algorithms for computing the b -function 1 Functional equation, P ( s ) f • f s = b f ( s ) · f s . 2 By definition, (Ann D n [ s ] ( f s ) + � f � ) ∩ C [ s ] = � b f ( s ) � . 3 Now find a system of generator of the annihilator and proceed with the elimination. Annihilator Elimination Oaku-Takayama (1997) Noro (2002) Brian¸ con-Maisonobe (2002) Andre-Levandovskyy-MM (2009) Levandovskyy (2008) J. Mart´ ın-Morales (jorge@unizar.es) Algorithms for Computing Bernstein-Sato Polynomials
Second part of the talk Partial solution: the checkRoot algorithm Applications: 1 Computation of b -functions via upper bounds. 2 Integral roots of b -functions. 3 Stratification associated with local b -functions without employing primary ideal decomposition. J. Mart´ ın-Morales (jorge@unizar.es) Algorithms for Computing Bernstein-Sato Polynomials
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