By Andreas Maletti

ISBN-10: 3319230204

ISBN-13: 9783319230207

ISBN-10: 3319230212

ISBN-13: 9783319230214

This e-book constitutes the refereed lawsuits of the sixth overseas convention on Algebraic Informatics, CAI 2015, held in Stuttgart, Germany, in September 2015.

The 15 revised complete papers awarded have been conscientiously reviewed and chosen from 25 submissions. The papers disguise subject matters akin to facts versions and coding conception; basic features of cryptography and defense; algebraic and stochastic types of computing; common sense and software modelling.

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Extra info for Algebraic Informatics: 6th International Conference, CAI 2015, Stuttgart, Germany, September 1-4, 2015. Proceedings

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A reduction from WordEquations to Hilbert 10 is now straightforward. For example, the equation abX = Y ba is solvable if and only if the following Diophantine system in unknowns X1 , . . , Y4 is solvable over integers: ( 11 01 ) · ( 10 11 ) · X1 X2 X3 X4 Y1 Y2 Y3 Y4 = · ( 10 11 ) · ( 11 01 ) X1 X4 − X2 X3 = 1 Y1 Y4 − Y2 Y3 = 1 Xi ≥ 0 & Yi ≥ 0 for 1 ≤ i ≤ 4 The reduction of a Diophantine system to a single Diophantine equation is classic. It is based on the fact that every natural number can be written as a sum of four squares.

Fn = {g1 f1 + · · · + gn fn | g1 , . . , gn ∈ C[X ±1 ]} An Algebraic Geometric Approach to Multidimensional Words 37 be the Laurent polynomial ideal they generate. Note that in this notation we let all involved polynomials be Laurent so that this is not a polynomial ideal. For Laurent polynomials f (X) and g(X), we denote f ≡ g mod f1 , . . , fn if and only if f (X) − g(X) ∈ f1 , . . , fn . A (Laurent) polynomial a(X) is called a line (Laurent) polynomial if the support supp(a) defined by (2) contains at least two points and all the points of the support lie on a single line.

32 J. Kari and M. Szabados Configuration c has low complexity with respect to a finite D ⊆ Zd if | PattD (c)| ≤ |D|, (3) and we say that c has low complexity if (3) is satisfied for some finite D. We say that a Laurent polynomial f (X) annihilates configuration c(X) if f (X)c(X) = 0. The following lemma guarantees that each low complexity configuration is annihilated by some non-zero Laurent polynomial, and hence also by a non-zero proper polynomial. Lemma 1. Let R be a field or R = Z. Let c(X) ∈ R[[X ±1 ]] be a configuration and D ⊂ Zd a finite set such that | PattD (c)| ≤ |D|.

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Algebraic Informatics: 6th International Conference, CAI 2015, Stuttgart, Germany, September 1-4, 2015. Proceedings by Andreas Maletti


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