Integral Bases for P-Recursive Sequences Lixin Du Chinese Academy - PowerPoint PPT Presentation

Integral Bases for P-Recursive Sequences Lixin Du Chinese Academy of Sciences Johannes Kepler University Linz ISSAC 2020, Kalamata, Greece July 20-23, 2020 , 1/18

Integral Bases for P-Recursive Sequences Lixin Du Chinese Academy of Sciences Johannes Kepler University Linz ISSAC 2020, Kalamata, Greece July 20-23, 2020 joint work with S. Chen, M. Kauers, and T. Verron , 1/18

Integral functions: algebraic case Notation. C : a field of characteristic zero (e.g. Q , C ). C ( x ) : the field of rational functions in x . M ∈ C [ x , y ] : irreducible over C ( x ) with r = deg y ( M ) . K = C ( x )[ y ] / � M � ∼ = C ( x )( β ) : the algebraic function field. � � a 0 + a 1 β + ··· + a r − 1 β r − 1 | a i ∈ C ( x ) K = , 2/18

Integral functions: algebraic case Notation. C : a field of characteristic zero (e.g. Q , C ). C ( x ) : the field of rational functions in x . M ∈ C [ x , y ] : irreducible over C ( x ) with r = deg y ( M ) . K = C ( x )[ y ] / � M � ∼ = C ( x )( β ) : the algebraic function field. � � a 0 + a 1 β + ··· + a r − 1 β r − 1 | a i ∈ C ( x ) K = Definition. A function f ∈ K is called integral over C [ x ] if f d + p d − 1 f d − 1 + ··· + p 0 = 0 with p i ∈ C [ x ] . , 2/18

Integral bases: algebraic case Definition. The integral closure of C [ x ] in K is defined as O K := { f ∈ K | f is integral over C [ x ] } . , 3/18

Integral bases: algebraic case Definition. The integral closure of C [ x ] in K is defined as O K := { f ∈ K | f is integral over C [ x ] } . Theorem. O K is a free C [ x ] -module of rank r = [ K : C ( x )] . , 3/18

Integral bases: algebraic case Definition. The integral closure of C [ x ] in K is defined as O K := { f ∈ K | f is integral over C [ x ] } . Theorem. O K is a free C [ x ] -module of rank r = [ K : C ( x )] . Definition. An integral basis is a basis for O K as a C [ x ] -module. , 3/18

Integral bases: algebraic case Definition. The integral closure of C [ x ] in K is defined as O K := { f ∈ K | f is integral over C [ x ] } . Theorem. O K is a free C [ x ] -module of rank r = [ K : C ( x )] . Definition. An integral basis is a basis for O K as a C [ x ] -module. √ 3 Example. K = C ( x )( β ) with β = x 2 . � x β 2 � 1 , β , 1 is an integral basis. , 3/18

Integral bases: algebraic case Definition. The integral closure of C [ x ] in K is defined as O K := { f ∈ K | f is integral over C [ x ] } . Theorem. O K is a free C [ x ] -module of rank r = [ K : C ( x )] . Definition. An integral basis is a basis for O K as a C [ x ] -module. Question. Given an irreducible polynomial M ∈ C ( x )[ y ] , how to find an integral basis for K = C ( x )[ y ] / � M � ? , 3/18

Integral bases: algebraic case Definition. The integral closure of C [ x ] in K is defined as O K := { f ∈ K | f is integral over C [ x ] } . Theorem. O K is a free C [ x ] -module of rank r = [ K : C ( x )] . Definition. An integral basis is a basis for O K as a C [ x ] -module. Question. Given an irreducible polynomial M ∈ C ( x )[ y ] , how to find an integral basis for K = C ( x )[ y ] / � M � ? Answer. van Hoeij’s algorithm, etc. , 3/18

Computation of integral bases: van Hoeij’s algorithm Input. M ∈ C [ x , y ] monic irreducible over C ( x ) with r = deg y ( M ) ; output. an integral basis { B 0 , ··· , B r − 1 } . 1. Start with ( B 0 ,..., B r − 1 ) := ( 1 , β ,..., β r − 1 ) . 2. For d ∈ { 0 , 1 ,..., r − 1 } 3. While there exist a 0 ,..., a d − 1 ∈ C [ x ] such that A = a 0 B 0 + ··· + a d − 1 B d − 1 + B d p ( x ) is integral and p ( x ) ∈ C [ x ] \ C ; replace B d by A . 4. Return B 0 ,..., B r − 1 . , 4/18

Computation of integral bases: van Hoeij’s algorithm 16 x 3 + 2 x 4 )− x 3 y −( 2 x + 1 ) y 2 + y 3 . Example. M = ( 25

Computation of integral bases: van Hoeij’s algorithm 16 x 3 + 2 x 4 )− x 3 y −( 2 x + 1 ) y 2 + y 3 . Example. M = ( 25 integral closure O K

Computation of integral bases: van Hoeij’s algorithm 16 x 3 + 2 x 4 )− x 3 y −( 2 x + 1 ) y 2 + y 3 . Example. M = ( 25 integral closure O K O 0 C [ x ]+ C [ x ] β + C [ x ] β 2

Computation of integral bases: van Hoeij’s algorithm 16 x 3 + 2 x 4 )− x 3 y −( 2 x + 1 ) y 2 + y 3 . Example. M = ( 25 integral closure O K B 2 := − β + β 2 α = 0 , O 1 x O 0 C [ x ]+ C [ x ] β + C [ x ] β 2

Computation of integral bases: van Hoeij’s algorithm 16 x 3 + 2 x 4 )− x 3 y −( 2 x + 1 ) y 2 + y 3 . Example. M = ( 25 integral closure O K B d := a 0 B 0 + ··· a d − 1 B d − 1 + B d α ∈ C , O n x − α ··· B 2 := − β + β 2 α = 0 , O 1 x O 0 C [ x ]+ C [ x ] β + C [ x ] β 2 , 5/18

Computation of integral bases: van Hoeij’s algorithm 16 x 3 + 2 x 4 )− x 3 y −( 2 x + 1 ) y 2 + y 3 . Example. M = ( 25 integral closure O K B d := a 0 B 0 + ··· a d − 1 B d − 1 + B d α ∈ C , O n x − α ··· B 2 := − β + β 2 α = 0 , O 1 x O 0 C [ x ]+ C [ x ] β + C [ x ] β 2 { 1 , β , 1 x (− β + β 2 ) } is an integral basis of C ( x )[ y ] / � M � . , 5/18

Integral bases: general framework Definition. Let k be a field. The map ν : k → Z ∪ { ∞ } is called a valuation if for all a , b ∈ k ν ( a ) = ∞ iff a = 0 ; ν ( ab ) = ν ( a )+ ν ( b ) ; ν ( a + b ) ≥ min { ν ( a ) , ν ( b ) } . , 6/18

Integral bases: general framework Definition. Let k be a field. The map ν : k → Z ∪ { ∞ } is called a valuation if for all a , b ∈ k ν ( a ) = ∞ iff a = 0 ; ν ( ab ) = ν ( a )+ ν ( b ) ; ν ( a + b ) ≥ min { ν ( a ) , ν ( b ) } . The pair ( k , ν ) is called a valued field. , 6/18

Integral bases: general framework Definition. Let k be a field. The map ν : k → Z ∪ { ∞ } is called a valuation if for all a , b ∈ k ν ( a ) = ∞ iff a = 0 ; ν ( ab ) = ν ( a )+ ν ( b ) ; ν ( a + b ) ≥ min { ν ( a ) , ν ( b ) } . The pair ( k , ν ) is called a valued field. The valuation ring O ( k , ν ) of k is defined as O ( k , ν ) = { a ∈ k | ν ( a ) ≥ 0 } . , 6/18

Integral bases: general framework Definition. Let k be a field. The map ν : k → Z ∪ { ∞ } is called a valuation if for all a , b ∈ k ν ( a ) = ∞ iff a = 0 ; ν ( ab ) = ν ( a )+ ν ( b ) ; ν ( a + b ) ≥ min { ν ( a ) , ν ( b ) } . The pair ( k , ν ) is called a valued field. The valuation ring O ( k , ν ) of k is defined as O ( k , ν ) = { a ∈ k | ν ( a ) ≥ 0 } . Example. For a nonzero f ∈ C ( x ) , define ν z ( f ) = m if f = ( x − z ) m a where ( x − z ) ∤ a , b . b , 6/18

Integral bases: general framework Definition. Let k be a field. The map ν : k → Z ∪ { ∞ } is called a valuation if for all a , b ∈ k ν ( a ) = ∞ iff a = 0 ; ν ( ab ) = ν ( a )+ ν ( b ) ; ν ( a + b ) ≥ min { ν ( a ) , ν ( b ) } . The pair ( k , ν ) is called a valued field. The valuation ring O ( k , ν ) of k is defined as O ( k , ν ) = { a ∈ k | ν ( a ) ≥ 0 } . Example. ( C ( x ) , ν z ) is a valued field. , 6/18

Integral bases: general framework Definition. Let k be a field. The map ν : k → Z ∪ { ∞ } is called a valuation if for all a , b ∈ k ν ( a ) = ∞ iff a = 0 ; ν ( ab ) = ν ( a )+ ν ( b ) ; ν ( a + b ) ≥ min { ν ( a ) , ν ( b ) } . The pair ( k , ν ) is called a valued field. The valuation ring O ( k , ν ) of k is defined as O ( k , ν ) = { a ∈ k | ν ( a ) ≥ 0 } . Example. ( C ( x ) , ν z ) is a valued field. The corresponding valuation ring is � a � � C [ x ] x − z = b ∈ C ( x ) � ( x − z ) ∤ b . � , 6/18

Integral bases: general framework Definition. Let k be a field. The map ν : k → Z ∪ { ∞ } is called a valuation if for all a , b ∈ k ν ( a ) = ∞ iff a = 0 ; ν ( ab ) = ν ( a )+ ν ( b ) ; ν ( a + b ) ≥ min { ν ( a ) , ν ( b ) } . The pair ( k , ν ) is called a valued field. The valuation ring O ( k , ν ) of k is defined as O ( k , ν ) = { a ∈ k | ν ( a ) ≥ 0 } . Example. ( C ( x ) , ν z ) is a valued field. Fact. The valuation ring is integrally closed. , 6/18

Integral Bases: general framework Definition. Let V be a vector space over ( k , ν ) . The map val : V → Z ∪ { ∞ } is called a value function if for all B , B 1 , B 2 ∈ V and u ∈ k val ( B ) = ∞ iff B = 0 ; val ( u · B ) = ν ( u )+ val ( B ) ; val ( B 1 + B 2 ) ≥ min { val ( B 1 ) , val ( B 2 ) } . , 7/18

Integral Bases: general framework Definition. Let V be a vector space over ( k , ν ) . The map val : V → Z ∪ { ∞ } is called a value function if for all B , B 1 , B 2 ∈ V and u ∈ k val ( B ) = ∞ iff B = 0 ; val ( u · B ) = ν ( u )+ val ( B ) ; val ( B 1 + B 2 ) ≥ min { val ( B 1 ) , val ( B 2 ) } . The pair ( V , val ) is called a valued vector space. , 7/18

Integral Bases: general framework Definition. Let V be a vector space over ( k , ν ) . The map val : V → Z ∪ { ∞ } is called a value function if for all B , B 1 , B 2 ∈ V and u ∈ k val ( B ) = ∞ iff B = 0 ; val ( u · B ) = ν ( u )+ val ( B ) ; val ( B 1 + B 2 ) ≥ min { val ( B 1 ) , val ( B 2 ) } . The pair ( V , val ) is called a valued vector space. An element B of V is called integral if val ( B ) ≥ 0 . , 7/18