Generalisation of Canonical Number Systems Paul Surer (joint work - PowerPoint PPT Presentation

Generalisation of Canonical Number Systems Paul Surer (joint work with K. Scheicher, J. M. Thuswaldner and C. Van de Woestijne) Montanuniversität Leoben Department of Mathematics and Information Technology Chair of Mathematics and Statistics 8700 Leoben - AUSTRIA Prague, May 2008 Supported by FWF, project S9610 Paul Surer (MU Leoben) Generalised CNS Prague 2008 1 / 13

Introduction Canonical Number Systems Definition (Pethő, 1991) Let P ( x ) = x d + p d − 1 x d − 1 + · · · + p 1 x + p 0 , | p 0 | ≥ 2, R = Z [ x ] / ( P ) , X the image of x under the canonical epimorphism and N = [ 0 , | p 0 | ) ∩ Z . ( P , N ) is called a Canonical Number System (CNS) if each A ∈ R can be represented as k � a j X j , A = a j ∈ N . j = 0 Paul Surer (MU Leoben) Generalised CNS Prague 2008 2 / 13

Introduction Canonical Number Systems Definition (Pethő, 1991) Let P ( x ) = x d + p d − 1 x d − 1 + · · · + p 1 x + p 0 , | p 0 | ≥ 2, R = Z [ x ] / ( P ) , X the image of x under the canonical epimorphism and N = [ 0 , | p 0 | ) ∩ Z . ( P , N ) is called a Canonical Number System (CNS) if each A ∈ R can be represented as k � a j X j , A = a j ∈ N . j = 0 Examples of generalisations Akiyama et al. , 3 2 -problem, Paul Surer (MU Leoben) Generalised CNS Prague 2008 2 / 13

Introduction Canonical Number Systems Definition (Pethő, 1991) Let P ( x ) = x d + p d − 1 x d − 1 + · · · + p 1 x + p 0 , | p 0 | ≥ 2, R = Z [ x ] / ( P ) , X the image of x under the canonical epimorphism and N = [ 0 , | p 0 | ) ∩ Z . ( P , N ) is called a Canonical Number System (CNS) if each A ∈ R can be represented as k � a j X j , A = a j ∈ N . j = 0 Examples of generalisations Akiyama et al. , 3 2 -problem, Kovács, CNS rings. Paul Surer (MU Leoben) Generalised CNS Prague 2008 2 / 13

Digit systems Settings E a commutative ring with identity, Paul Surer (MU Leoben) Generalised CNS Prague 2008 3 / 13

Digit systems Settings E a commutative ring with identity, P ( X ) = p d X d + · · · + p 1 X + p 0 ∈ E [ X ] , d ≥ 1, Paul Surer (MU Leoben) Generalised CNS Prague 2008 3 / 13

Digit systems Settings E a commutative ring with identity, P ( X ) = p d X d + · · · + p 1 X + p 0 ∈ E [ X ] , d ≥ 1, p d no zero divisor, Paul Surer (MU Leoben) Generalised CNS Prague 2008 3 / 13

Digit systems Settings E a commutative ring with identity, P ( X ) = p d X d + · · · + p 1 X + p 0 ∈ E [ X ] , d ≥ 1, p d no zero divisor, p 0 neither a unit nor a zero divisor, E / p 0 finite, Paul Surer (MU Leoben) Generalised CNS Prague 2008 3 / 13

Digit systems Settings E a commutative ring with identity, P ( X ) = p d X d + · · · + p 1 X + p 0 ∈ E [ X ] , d ≥ 1, p d no zero divisor, p 0 neither a unit nor a zero divisor, E / p 0 finite, R = E [ X ] / ( P ) the residue class ring, Paul Surer (MU Leoben) Generalised CNS Prague 2008 3 / 13

Digit systems Settings E a commutative ring with identity, P ( X ) = p d X d + · · · + p 1 X + p 0 ∈ E [ X ] , d ≥ 1, p d no zero divisor, p 0 neither a unit nor a zero divisor, E / p 0 finite, R = E [ X ] / ( P ) the residue class ring, N ⊂ R a set of representatives of R / ( X ) , Paul Surer (MU Leoben) Generalised CNS Prague 2008 3 / 13

Digit systems Settings E a commutative ring with identity, P ( X ) = p d X d + · · · + p 1 X + p 0 ∈ E [ X ] , d ≥ 1, p d no zero divisor, p 0 neither a unit nor a zero divisor, E / p 0 finite, R = E [ X ] / ( P ) the residue class ring, N ⊂ R a set of representatives of R / ( X ) , we will identify X ∈ E [ X ] with X ∈ R , Paul Surer (MU Leoben) Generalised CNS Prague 2008 3 / 13

Digit systems Settings E a commutative ring with identity, P ( X ) = p d X d + · · · + p 1 X + p 0 ∈ E [ X ] , d ≥ 1, p d no zero divisor, p 0 neither a unit nor a zero divisor, E / p 0 finite, R = E [ X ] / ( P ) the residue class ring, N ⊂ R a set of representatives of R / ( X ) , we will identify X ∈ E [ X ] with X ∈ R , m N : R → N such that A ≡ m N ( A ) mod X , Paul Surer (MU Leoben) Generalised CNS Prague 2008 3 / 13

Digit systems Settings E a commutative ring with identity, P ( X ) = p d X d + · · · + p 1 X + p 0 ∈ E [ X ] , d ≥ 1, p d no zero divisor, p 0 neither a unit nor a zero divisor, E / p 0 finite, R = E [ X ] / ( P ) the residue class ring, N ⊂ R a set of representatives of R / ( X ) , we will identify X ∈ E [ X ] with X ∈ R , m N : R → N such that A ≡ m N ( A ) mod X , T P : R → R , A �→ A − m N ( A ) . X Paul Surer (MU Leoben) Generalised CNS Prague 2008 3 / 13

Digit systems Settings E a commutative ring with identity, P ( X ) = p d X d + · · · + p 1 X + p 0 ∈ E [ X ] , d ≥ 1, p d no zero divisor, p 0 neither a unit nor a zero divisor, E / p 0 finite, R = E [ X ] / ( P ) the residue class ring, N ⊂ R a set of representatives of R / ( X ) , we will identify X ∈ E [ X ] with X ∈ R , m N : R → N such that A ≡ m N ( A ) mod X , T P : R → R , A �→ A − m N ( A ) . X Paul Surer (MU Leoben) Generalised CNS Prague 2008 3 / 13

Digit systems Settings E a commutative ring with identity, P ( X ) = p d X d + · · · + p 1 X + p 0 ∈ E [ X ] , d ≥ 1, p d no zero divisor, p 0 neither a unit nor a zero divisor, E / p 0 finite, R = E [ X ] / ( P ) the residue class ring, N ⊂ R a set of representatives of R / ( X ) , we will identify X ∈ E [ X ] with X ∈ R , m N : R → N such that A ≡ m N ( A ) mod X , T P : R → R , A �→ A − m N ( A ) . X Note that R / ( X ) ∼ = E / p 0 . Paul Surer (MU Leoben) Generalised CNS Prague 2008 3 / 13

Digit systems X -ary representation Definition We call the triple ( R , X , N ) a digit system in R . Paul Surer (MU Leoben) Generalised CNS Prague 2008 4 / 13

Digit systems X -ary representation Definition We call the triple ( R , X , N ) a digit system in R . Definition P ( A ))) n ∈ N ⊂ N ∞ the X -ary For an A ∈ R we call X P ( A ) := ( m N ( T n representation. X P ( A ) is called periodic if it ends up periodically. ( R , X , N ) has the periodic representation property if X P ( A ) is periodic for all A ∈ R . Paul Surer (MU Leoben) Generalised CNS Prague 2008 4 / 13

Digit systems X -ary expansion Definition Let A ∈ R and X P ( A ) = ( a n ) n ∈ N . If there exists an l ∈ N such that l � a j X j A = j = 0 we call this sum the finite X -ary expansion of A . ( R , X , N ) is said to have the finite expansion property if each A ∈ R has a finite X -ary expansion. Paul Surer (MU Leoben) Generalised CNS Prague 2008 5 / 13

Properties of digit systems General results Proposition A ∈ R has a finite X -ary expansion if and only if T n P ( A ) = 0 for some n ∈ N . Paul Surer (MU Leoben) Generalised CNS Prague 2008 6 / 13

Properties of digit systems General results Proposition A ∈ R has a finite X -ary expansion if and only if T n P ( A ) = 0 for some n ∈ N . Proposition The finite expansion property implies the periodic representation property. Paul Surer (MU Leoben) Generalised CNS Prague 2008 6 / 13

Properties of digit systems Denote by P ( R , X , N ) the set of purely periodic points: P ( R , X , N ) = { A ∈ R| ∃ n ≥ 1 : T n P ( A ) = A } . Paul Surer (MU Leoben) Generalised CNS Prague 2008 7 / 13

Properties of digit systems Denote by P ( R , X , N ) the set of purely periodic points: P ( R , X , N ) = { A ∈ R| ∃ n ≥ 1 : T n P ( A ) = A } . Theorem ( R , X , N ) has the finite expansion property if and only if ( R , X , N ) has the periodic representation property, 0 ∈ P ( R , X , N ) and |P ( R , X , N ) / T P | = 1 . Paul Surer (MU Leoben) Generalised CNS Prague 2008 7 / 13

Properties of digit systems Denote by P ( R , X , N ) the set of purely periodic points: P ( R , X , N ) = { A ∈ R| ∃ n ≥ 1 : T n P ( A ) = A } . Theorem ( R , X , N ) has the finite expansion property if and only if ( R , X , N ) has the periodic representation property, 0 ∈ P ( R , X , N ) and |P ( R , X , N ) / T P | = 1 . Theorem (Composition Theorem) Let ( R 1 , X , N 1 ) , ( R 2 , X , N 2 ) induced by the polynomials P 1 ∈ E [ x ] and P 2 ∈ E [ x ] and suppose both of them to have the finite expansion property. If ( |P ( R 1 , X , N 1 ) | , |P ( R 2 , X , N 2 ) | ) = 1 . Then ( R , X , N ) has the finite expansion property for R = E [ X ] / ( P 1 P 2 ) and N = { d + eP 1 | d ∈ N 1 , e ∈ N 2 } . Paul Surer (MU Leoben) Generalised CNS Prague 2008 7 / 13

Main theorem and consequences The P -lattice Definition Let w 0 = p d , w k = Xw k − 1 + p d − k for k = 1 , . . . , d − 1. We call the E -submodule of R generated by the w i the P -lattice of R and denote it by Λ P ( R ) . Paul Surer (MU Leoben) Generalised CNS Prague 2008 8 / 13

Main theorem and consequences The P -lattice Definition Let w 0 = p d , w k = Xw k − 1 + p d − k for k = 1 , . . . , d − 1. We call the E -submodule of R generated by the w i the P -lattice of R and denote it by Λ P ( R ) . For p d = ± 1 we have Λ P ( R ) = R . This is definitely NOT true for p d � = ± 1. Paul Surer (MU Leoben) Generalised CNS Prague 2008 8 / 13