Constructive Arithmetics in Ore Localizations with Enough - PowerPoint PPT Presentation

Constructive Arithmetics in Ore Localizations with Enough Commutativity Johannes Hoffmann, Viktor Levandovskyy 1 RWTH Aachen University, Germany ISSAC 2018, New York 1 supported by DFG Transregio 195 Johannes Hoffmann (RWTH Aachen) Constructive Ore Arithmetics ISSAC 2018, New York 1 / 31

Content Motivation 1 The intersection problem in polynomial algebras 2 The closure problem 3 Johannes Hoffmann (RWTH Aachen) Constructive Ore Arithmetics ISSAC 2018, New York 2 / 31

Content Motivation 1 The intersection problem in polynomial algebras 2 The closure problem 3 Johannes Hoffmann (RWTH Aachen) Constructive Ore Arithmetics ISSAC 2018, New York 3 / 31

Localization of commutative domains Definition A subset S of a ring R is called multiplicative set if 0 / ∈ S , 1 ∈ S and S is multiplicatively closed , that is, ∀ s , t ∈ S : s · t ∈ S . Notation: [ S ] := the smallest multiplicative superset of S . Johannes Hoffmann (RWTH Aachen) Constructive Ore Arithmetics ISSAC 2018, New York 4 / 31

Localization of commutative domains Definition A subset S of a ring R is called multiplicative set if 0 / ∈ S , 1 ∈ S and S is multiplicatively closed , that is, ∀ s , t ∈ S : s · t ∈ S . Notation: [ S ] := the smallest multiplicative superset of S . Theorem (Classical) Let S be a multiplicative set in a commutative domain R . Then � r � � � S − 1 R := s − 1 r | ( s , r ) ∈ S × R s | r ∈ R , s ∈ S = is a commutative domain (with the usual addition and multiplication of fractions). Johannes Hoffmann (RWTH Aachen) Constructive Ore Arithmetics ISSAC 2018, New York 4 / 31

Commutative examples Classical localizations Let R be a commutative domain. � � p = ( R \ { 0 } ) − 1 R Quot( R ) := q | p , q ∈ R , q � = 0 � � p = ( R \ p ) − 1 R , p prime ideal of R R p := q | p , q ∈ R , q / ∈ p � p � = [ f ] − 1 R , f ∈ R \ { 0 } R f := f k | p ∈ R , k ∈ N 0 R p Quot( R ) R R f Johannes Hoffmann (RWTH Aachen) Constructive Ore Arithmetics ISSAC 2018, New York 5 / 31

The hierarchy of Ore localizations: localization of. . . commutative domains

The hierarchy of Ore localizations: localization of. . . commutative commutative rings commutative domains

The hierarchy of Ore localizations: localization of. . . commutative domains arbitrary domains commutative rings commutative domains

The hierarchy of Ore localizations: localization of. . . arbitrary rings commutative domains arbitrary domains commutative rings commutative domains Johannes Hoffmann (RWTH Aachen) Constructive Ore Arithmetics ISSAC 2018, New York 6 / 31

Left Ore sets, left denominator sets Definition Let S be a subset of a ring R . S satisfies the left Ore condition in R if ∀ s ∈ S , r ∈ R ∃ ˜ s ∈ S , ˜ r ∈ R : sr = ˜ ˜ rs . Johannes Hoffmann (RWTH Aachen) Constructive Ore Arithmetics ISSAC 2018, New York 7 / 31

Left Ore sets, left denominator sets Definition Let S be a subset of a ring R . S satisfies the left Ore condition in R if ∀ s ∈ S , r ∈ R ∃ ˜ s ∈ S , ˜ r ∈ R : sr = ˜ ˜ rs . Left Ore set = multiplicative set + left Ore condition Johannes Hoffmann (RWTH Aachen) Constructive Ore Arithmetics ISSAC 2018, New York 7 / 31

Left Ore sets, left denominator sets Definition Let S be a subset of a ring R . S satisfies the left Ore condition in R if ∀ s ∈ S , r ∈ R ∃ ˜ s ∈ S , ˜ r ∈ R : sr = ˜ ˜ rs . Left Ore set = multiplicative set + left Ore condition S is left reversible in R if ∀ s ∈ S , r ∈ R : ⇒ ∃ ˜ s ∈ S : ˜ rs = 0 sr = 0 . Johannes Hoffmann (RWTH Aachen) Constructive Ore Arithmetics ISSAC 2018, New York 7 / 31

Left Ore sets, left denominator sets Definition Let S be a subset of a ring R . S satisfies the left Ore condition in R if ∀ s ∈ S , r ∈ R ∃ ˜ s ∈ S , ˜ r ∈ R : sr = ˜ ˜ rs . Left Ore set = multiplicative set + left Ore condition S is left reversible in R if ∀ s ∈ S , r ∈ R : ⇒ ∃ ˜ s ∈ S : ˜ rs = 0 sr = 0 . Left denominator set = left Ore set + left reversibility Johannes Hoffmann (RWTH Aachen) Constructive Ore Arithmetics ISSAC 2018, New York 7 / 31

Left Ore sets, left denominator sets Definition Let S be a subset of a ring R . S satisfies the left Ore condition in R if ∀ s ∈ S , r ∈ R ∃ ˜ s ∈ S , ˜ r ∈ R : sr = ˜ ˜ rs . Left Ore set = multiplicative set + left Ore condition S is left reversible in R if ∀ s ∈ S , r ∈ R : ⇒ ∃ ˜ s ∈ S : ˜ rs = 0 sr = 0 . Left denominator set = left Ore set + left reversibility Consequences of the left Ore condition on S in R Finite collections of elements from S have common left multiples in S . Any right fraction rs − 1 can be rewritten as a left fraction ˜ s − 1 ˜ r . Johannes Hoffmann (RWTH Aachen) Constructive Ore Arithmetics ISSAC 2018, New York 7 / 31

Construction of the left Ore localization Theorem (Ore, 1931) The following is an equivalence relation on S × R : ( s 1 , r 1 ) ∼ ( s 2 , r 2 ) ⇔ ∃ ˜ s ∈ S , ˜ r ∈ R : ˜ ss 2 = ˜ rs 1 and ˜ sr 2 = ˜ rr 1 Johannes Hoffmann (RWTH Aachen) Constructive Ore Arithmetics ISSAC 2018, New York 8 / 31

Construction of the left Ore localization Theorem (Ore, 1931) The following is an equivalence relation on S × R : ( s 1 , r 1 ) ∼ ( s 2 , r 2 ) ⇔ ∃ ˜ s ∈ S , ˜ r ∈ R : ˜ ss 2 = ˜ rs 1 and ˜ sr 2 = ˜ rr 1 S − 1 R := ( S × R / ∼ , + , · ) is a ring via + : S − 1 R × S − 1 R → S − 1 R , ( s 1 , r 1 ) + ( s 2 , r 2 ) := (˜ ss 1 , ˜ sr 1 + ˜ rr 2 ) , s ∈ S and ˜ r ∈ R satisfy ˜ where ˜ ss 1 = ˜ rs 2 , and · : S − 1 R × S − 1 R → S − 1 R , ( s 1 , r 1 ) · ( s 2 , r 2 ) := (˜ ss 1 , ˜ rr 2 ) , s ∈ S and ˜ r ∈ R satisfy ˜ where ˜ sr 1 = ˜ rs 2 . Johannes Hoffmann (RWTH Aachen) Constructive Ore Arithmetics ISSAC 2018, New York 8 / 31

Partial classification of Ore localizations Definition Let K be a field and R a K -algebra, S a left denominator set in R . Johannes Hoffmann (RWTH Aachen) Constructive Ore Arithmetics ISSAC 2018, New York 9 / 31

Partial classification of Ore localizations Definition Let K be a field and R a K -algebra, S a left denominator set in R . Monoidal localization: S is generated as a monoid by countably many elements Example: [ f 1 , . . . , f k ] − 1 K [ x ] , f i ∈ K [ x ] \ { 0 } Johannes Hoffmann (RWTH Aachen) Constructive Ore Arithmetics ISSAC 2018, New York 9 / 31

Partial classification of Ore localizations Definition Let K be a field and R a K -algebra, S a left denominator set in R . Monoidal localization: S is generated as a monoid by countably many elements Example: [ f 1 , . . . , f k ] − 1 K [ x ] , f i ∈ K [ x ] \ { 0 } Geometric localization: Let n ∈ N , K [ x ] := K [ x 1 , . . . , x n ] , J an ideal in K [ x ] , p a prime ideal in K [ x ] / J and S = ( K [ x ] / J ) \ p Example: K [ x ] p Johannes Hoffmann (RWTH Aachen) Constructive Ore Arithmetics ISSAC 2018, New York 9 / 31

Partial classification of Ore localizations Definition Let K be a field and R a K -algebra, S a left denominator set in R . Monoidal localization: S is generated as a monoid by countably many elements Example: [ f 1 , . . . , f k ] − 1 K [ x ] , f i ∈ K [ x ] \ { 0 } Geometric localization: Let n ∈ N , K [ x ] := K [ x 1 , . . . , x n ] , J an ideal in K [ x ] , p a prime ideal in K [ x ] / J and S = ( K [ x ] / J ) \ p Example: K [ x ] p Rational localization: T ⊆ R is a K -subalgebra, S = T \ { 0 } Special case: R is generated by a set X of variables and T is generated by a subset of X ⇒ S − 1 R is essential rational Example: ( K [ x ] \ { 0 } ) − 1 K [ x , y ] ∼ = K ( x )[ y ] Johannes Hoffmann (RWTH Aachen) Constructive Ore Arithmetics ISSAC 2018, New York 9 / 31

Previously on ISSAC’17 Setup: a left Ore set S in a (not necessarily commutative) domain R . Goal: provide algorithms for basic arithmetic in S − 1 R . Restrictions: R is a G -algebra, S belongs to one of the types above a . Key problem: intersection of left ideal with submonoid Result: library olga.lib for Singular:Plural Johannes Hoffmann and Viktor Levandovskyy. A Constructive Approach to Arithmetics in Ore Localizations. In Proc. ISSAC’17 , pages 197–204. ACM Press, 2017. a Note that further computability restrictions apply. Johannes Hoffmann (RWTH Aachen) Constructive Ore Arithmetics ISSAC 2018, New York 10 / 31

Addressing the key problem The intersection problem Let S be a left denominator set in a ring R and I a left ideal in R . The intersection problem is to decide whether I ∩ S = ∅ and to compute an element contained in this intersection when the answer is negative. Johannes Hoffmann (RWTH Aachen) Constructive Ore Arithmetics ISSAC 2018, New York 11 / 31

Addressing the key problem The intersection problem Let S be a left denominator set in a ring R and I a left ideal in R . The intersection problem is to decide whether I ∩ S = ∅ and to compute an element contained in this intersection when the answer is negative. Recent result (Posur, 2018) The intersection problem is a main ingredient for solving linear systems over localizations of commutative rings. Johannes Hoffmann (RWTH Aachen) Constructive Ore Arithmetics ISSAC 2018, New York 11 / 31