In this paper, the relationship between the decomposition of (linear, time-invariant, differential) behaviors and the solvability of certain two-sided diophantine equations is explored. The possibility of expressing a behavior as the sum of two sub-behaviors, endowed with a finite dimensional (and hence autonomous) intersection, one of which is a priori chosen, proves to be related to the solvability of a particular two-sided diophantine equation. In particular, the existence of a direct sum decomposition is equivalent to the solvability of a two-sided Bezout equation, and hence to the internal skew-primeness of a suitable matrix pair.

### Behavior decompositions and two-sided diophantine equations

#### Abstract

In this paper, the relationship between the decomposition of (linear, time-invariant, differential) behaviors and the solvability of certain two-sided diophantine equations is explored. The possibility of expressing a behavior as the sum of two sub-behaviors, endowed with a finite dimensional (and hence autonomous) intersection, one of which is a priori chosen, proves to be related to the solvability of a particular two-sided diophantine equation. In particular, the existence of a direct sum decomposition is equivalent to the solvability of a two-sided Bezout equation, and hence to the internal skew-primeness of a suitable matrix pair.
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2001
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11577/2473144`
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