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By Andreas Maletti

This booklet constitutes the refereed court cases of the sixth overseas convention on Algebraic Informatics, CAI 2015, held in Stuttgart, Germany, in September 2015.

The 15 revised complete papers provided have been rigorously reviewed and chosen from 25 submissions. The papers disguise themes comparable to facts versions and coding thought; basic elements of cryptography and safety; algebraic and stochastic versions of computing; common sense and application modelling.

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

Example text

But then f (X) = a0 X 0 + a1 X u + · · · + an X nu is in a, b . If f (X) is a monomial then 1 ∈ a, b and hence a, b contains all Laurent polynomials. Otherwise f (X) has at least two non-zero coefficients and it is then a line Laurent polynomial in direction u. 38 J. Kari and M. Szabados The next lemma states that one-directed configurations in different directions are linearly independent. Lemma 5. Let c1 (X), . . , cn (X) be two-dimensional configurations that are one-directed and pairwise non-parallel.

Vd xv11 . . xvdd . vd =−∞ As usual, we abbreviate the vector (x1 , . . , xd ) of variables as X, and write monomial xv11 . . xvdd as X v for v = (v1 , . . , vd ) ∈ Zd . Configuration c can now be expressed compactly as cv X v . c(X) = (1) v∈Zd Usually we let A ⊆ Z so that configurations are power series with integer coefficients, but to use Nullstellensatz we need an algebraically closed field, so that An Algebraic Geometric Approach to Multidimensional Words 31 frequently we consider multivariate power series and polynomials over C.

By (6) this means (j2 − j1 )h = j2 nα − j1 nα , so that h is a rational number and cannot hence be equal to irrational nα. But then, using (6) again, limj→∞ pjn = ±∞ so that p(x) cannot be finitary, a contradiction. Now it is clear that (5) is a non-trivial linear dependency among one-directed configurations in pairwise non-parallel directions. This is impossible by Lemma 5 so (5) cannot hold. We have proved the following result: Theorem 4. Let α > 0 be irrational. The two-dimensional configuration s over the binary alphabet {0, 1} defined by sij = (i + j)α − iα − jα is a sum of three periodic integral configurations but not a sum of finitely many finitary periodic configurations.

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