Elasticity in Numerical Semigroup Rings and Power Series Rings Paul - PowerPoint PPT Presentation

Background Irreducibles K finite K infinite Elasticity in Numerical Semigroup Rings and Power Series Rings Paul Baginski Fairfield University INdAM International Meeting on Numerical Semigroups Cortona, Italy September 11, 2014 Paul Baginski Fairfield University Elasticity in Numerical Semigroup Rings and Power Series Rings

Background Irreducibles K finite K infinite Work building off undergraduate research by Chris Crutchfield, K. Grace Kennedy, Matthew Wright in the 2005 Trinity NSF Research Experience for Undergraduates. Paul Baginski Fairfield University Elasticity in Numerical Semigroup Rings and Power Series Rings

Background Irreducibles K finite K infinite Let S = � n 1 , . . . , n b � be a numerical semigroup, K a field. The semigroup ring of S over K is the subring of K [ x ] given by � � n � � � a i x i , where a i � = 0 ⇒ i ∈ S K [ S ] := f ∈ K [ x ] � f = � � i =0 The semigroup power series ring of S over K is the subring of K � x � given by � � � ∞ � � a i x i , where a i � = 0 ⇒ i ∈ S K � S � := f ∈ K � x � � f = � � i =0 Paul Baginski Fairfield University Elasticity in Numerical Semigroup Rings and Power Series Rings

Background Irreducibles K finite K infinite An element a ∈ K [ S ] • \ K [ S ] × is irreducible if a � = bc for any b , c ∈ K [ S ] \ K [ S ] × . Similarly, a ∈ K � S � • \ K � S � × is irreducible if a � = bc for any b , c ∈ K � S � \ K � S � × . We want to see how nonzero nonunits of K [ S ] (or K � S � ) factor as a product of irreducibles (up to associates). Paul Baginski Fairfield University Elasticity in Numerical Semigroup Rings and Power Series Rings

Background Irreducibles K finite K infinite Examples 1. Work in K [ x 2 , x 3 ]. x 6 = ( x 2 ) 3 = ( x 3 ) 2 S is isomorphic to the submonoid X = { x n | n ∈ S } ⊂ K [ S ]. Up to associates, this submonoid is saturated in K [ S ], meaning all factorizations in K [ S ] already occur in X . So the factorization theory of S is present in K [ S ]. Paul Baginski Fairfield University Elasticity in Numerical Semigroup Rings and Power Series Rings

Background Irreducibles K finite K infinite Examples 2. Work in K [ x 2 , x 5 ] with char K � = 2. x 10 (1 − x 2 ) 2 = [ x 2 ] 5 [1 − x 2 ][1 − x 2 ] = [ x 5 ] 2 [1 − x 2 ][1 − x 2 ] Paul Baginski Fairfield University Elasticity in Numerical Semigroup Rings and Power Series Rings

Background Irreducibles K finite K infinite Examples 2. Work in K [ x 2 , x 5 ] with char K � = 2. x 10 (1 − x 2 ) 2 = [ x 2 ] 5 [1 − x 2 ][1 − x 2 ] = [ x 5 ] 2 [1 − x 2 ][1 − x 2 ] = [ x 5 (1 + x )][ x 5 (1 − x )][1 − x 2 ] = [ x 2 ][ x 4 (1 + x )][ x 4 (1 − x )][1 − x 2 ] Paul Baginski Fairfield University Elasticity in Numerical Semigroup Rings and Power Series Rings

Background Irreducibles K finite K infinite Examples 2. Work in K [ x 2 , x 5 ] with char K � = 2. x 10 (1 − x 2 ) 2 = [ x 2 ] 5 [1 − x 2 ][1 − x 2 ] = [ x 5 ] 2 [1 − x 2 ][1 − x 2 ] = [ x 5 (1 + x )][ x 5 (1 − x )][1 − x 2 ] = [ x 2 ][ x 4 (1 + x )][ x 4 (1 − x )][1 − x 2 ] = [ x 5 (1 + x )(1 + x )][ x 5 (1 − x )(1 − x )] = [ x 2 ][ x 4 (1 + x )(1 + x )][ x 4 (1 − x )(1 − x )] and variants of the above. So x 10 (1 − x 2 ) 2 factors as a product of 7 , 4 , 3 , and 2 irreducibles. (in char 2 the final two expressions have reducible terms) Paul Baginski Fairfield University Elasticity in Numerical Semigroup Rings and Power Series Rings

Background Irreducibles K finite K infinite We need: language to quantify “non-uniqueness” of factorization criteria to test for irreducibility description of the factorization behavior in K [ S ], globally and, if lucky, locally. Paul Baginski Fairfield University Elasticity in Numerical Semigroup Rings and Power Series Rings

Background Irreducibles K finite K infinite Factorization language H a commutative, cancellative, atomic monoid, such as K [ S ] • or K � S � • under multiplication. H × = { f ∈ H | f is a unit } A ( H ) = { f ∈ H \ H × | f is irreducible } (the atoms of H ) Paul Baginski Fairfield University Elasticity in Numerical Semigroup Rings and Power Series Rings

Background Irreducibles K finite K infinite Factorization language The set of lengths of f is L ( f ) = { n ∈ N 0 |∃ a 1 , . . . , a n ∈ A ( H ) f = a 1 · · · a n } with ℓ ( f ) = min L ( f ) and L ( f ) = max L ( f ). The elasticity of f is ρ ( f ) = L ( f ) ℓ ( f ) and the elasticity of H is ρ ( H ) = sup { ρ ( f ) | f ∈ H \ H × } . Notes: Paul Baginski Fairfield University Elasticity in Numerical Semigroup Rings and Power Series Rings

Background Irreducibles K finite K infinite Factorization language The set of lengths of f is L ( f ) = { n ∈ N 0 |∃ a 1 , . . . , a n ∈ A ( H ) f = a 1 · · · a n } with ℓ ( f ) = min L ( f ) and L ( f ) = max L ( f ). The elasticity of f is ρ ( f ) = L ( f ) ℓ ( f ) and the elasticity of H is ρ ( H ) = sup { ρ ( f ) | f ∈ H \ H × } . Notes: (1) ρ ( f ) ≤ ρ ( f n ) for all n ∈ N . Paul Baginski Fairfield University Elasticity in Numerical Semigroup Rings and Power Series Rings

Background Irreducibles K finite K infinite Factorization language The set of lengths of f is L ( f ) = { n ∈ N 0 |∃ a 1 , . . . , a n ∈ A ( H ) f = a 1 · · · a n } with ℓ ( f ) = min L ( f ) and L ( f ) = max L ( f ). The elasticity of f is ρ ( f ) = L ( f ) ℓ ( f ) and the elasticity of H is ρ ( H ) = sup { ρ ( f ) | f ∈ H \ H × } . Notes: (1) ρ ( f ) ≤ ρ ( f n ) for all n ∈ N . (2) While ρ ( f ) ∈ Q , we could have ρ ( H ) be irrational or ∞ . Paul Baginski Fairfield University Elasticity in Numerical Semigroup Rings and Power Series Rings

Background Irreducibles K finite K infinite Questions: 1 Can we write out L ( x ) for any x ? or L ( x )? or ℓ ( x ) 2 Can we compute ρ ( x )? Or ρ ( H )? 3 Which rationals between 1 and ρ ( H ) can occur as ρ ( x ) of some x ∈ H ? 4 If ρ ( H ) ∈ Q , is there an x ∈ H such that ρ ( x ) = ρ ( H )? Paul Baginski Fairfield University Elasticity in Numerical Semigroup Rings and Power Series Rings

Background Irreducibles K finite K infinite Questions: 1 Can we write out L ( x ) for any x ? or L ( x )? or ℓ ( x ) 2 Can we compute ρ ( x )? Or ρ ( H )? 3 Which rationals between 1 and ρ ( H ) can occur as ρ ( x ) of some x ∈ H ? 4 If ρ ( H ) ∈ Q , is there an x ∈ H such that ρ ( x ) = ρ ( H )? H is fully elastic if for all q ∈ Q ∩ [1 , ρ ( H )), there is x ∈ H with ρ ( x ) = q . H has accepted elasticity if there is x ∈ H such that ρ ( x ) = ρ ( H ) ∈ Q . Paul Baginski Fairfield University Elasticity in Numerical Semigroup Rings and Power Series Rings

Background Irreducibles K finite K infinite Irreducibles We have submonoids = ( K × + xK [ x ]) / K × K 0 [ x ] = { f ∈ K [ x ] | f (0) = 1 } ∼ K 0 [ S ] = { f ∈ K [ S ] | f (0) = 1 } = K 0 [ x ] ∩ K [ S ] If f ∈ K [ S ], then we can write f uniquely in K [ x ] as f = ux n g 1 · · · g k , where u ∈ K × and g 1 , . . . , g k ∈ A ( K 0 [ x ]). We can immediately conclude n ∈ S . Suppose n = 0. What is special about g 1 , . . . , g k ? Paul Baginski Fairfield University Elasticity in Numerical Semigroup Rings and Power Series Rings

Background Irreducibles K finite K infinite Associated group Consider = ( K × + xK � x � ) / K × K 0 � x � = { f ∈ K � x � | f (0) = 1 } ∼ K 0 � S � = { f ∈ K � S � | f (0) = 1 } = K 0 � x � ∩ K � S � Clear that K � x � × = K × K 0 � x � . Paul Baginski Fairfield University Elasticity in Numerical Semigroup Rings and Power Series Rings

Background Irreducibles K finite K infinite Associated group Consider = ( K × + xK � x � ) / K × K 0 � x � = { f ∈ K � x � | f (0) = 1 } ∼ K 0 � S � = { f ∈ K � S � | f (0) = 1 } = K 0 � x � ∩ K � S � Clear that K � x � × = K × K 0 � x � . Less obvious: K 0 � S � is a group and K � S � × = K × K 0 � S � . Now set G S ( K ) = K 0 � x � / K 0 � S � Paul Baginski Fairfield University Elasticity in Numerical Semigroup Rings and Power Series Rings

Background Irreducibles K finite K infinite Associated group Consider = ( K × + xK � x � ) / K × K 0 � x � = { f ∈ K � x � | f (0) = 1 } ∼ K 0 � S � = { f ∈ K � S � | f (0) = 1 } = K 0 � x � ∩ K � S � Clear that K � x � × = K × K 0 � x � . Less obvious: K 0 � S � is a group and K � S � × = K × K 0 � S � . Now set G S ( K ) = K 0 � x � / K 0 � S � We have a homomorphism K 0 [ x ] → G S ( K ) where g ∈ K 0 [ x ] �→ g ∈ G S ( K ). Take g 1 , . . . , g k ∈ A ( K 0 [ x ]). If g 1 · · · g k ∈ K 0 [ S ] then g 1 · · · g k = g 1 · · · g k = 1 Paul Baginski Fairfield University Elasticity in Numerical Semigroup Rings and Power Series Rings