Ricordo di Alberto Tognoli
Divisors in global analytic sets
We prove that any divisor of a global analytic set has a generic equation, that is, there is an analytic function vanishing on with multiplicity one along each irreducible component of . We also prove that there are functions with arbitrary multiplicities along . The main result states that if is pure dimensional, is locally principal, is not connected and represents the zero class in then the divisor is globally principal.
Sull'insieme di non coerenza di un insieme analitico reale
In this short paper we show that the set of points in which a real analytic space is not coherent may be not analytic. (In [2] it is proved that it is always semianalytic).
Sulla non validità di un teorema di approssimazione
The following theorem is true: if is open in , is a coherent real analytic set, and is a function such that is analytic, then it is possible to approximate (together with its derivatives) by analytic functions such that . In this paper we prove that this result is not true unless is coherent (with the reduced structure).
On the finiteness of Pythagoras numbers of real meromorphic functions
We consider the 17th Hilbert Problem for global real analytic functions in a modified form that involves infinite sums of squares. Then we prove a local-global principle for a real global analytic function to be a sum of squares of global real meromorphic functions. We deduce that an affirmative solution to the 17th Hilbert Problem for global real analytic functions implies the finiteness of the Pythagoras number of the field of global real meromorphic functions, hence that of the field of real...
On the pythagoras numbers of real analytic surfaces
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