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[06] Sums of squares of linear forms: the quaternions approach.
With J.F. Fernando and C. Scheiderer.

Abstract: Let A be the polynomial ring in one single variable over a field k. We discuss the number of squares needed to represent sums of squares of linear forms with coefficients in the ring A. We use quaternions to obtain bounds when the Pythagoras number of A is not bigger than 4. This provides bounds for the Pythagoras number of algebraic curves and algebroid surfaces over the field k. (Click for PDF.) This was the starting step for the complete results in Papers [06].

Keywords: Pythagoras number, Pfister bound, quaternions.

Mathematical Subject Classification: 14P99, 11E25.


[04b] On the finiteness of Pythagoras numbers of real meromorphic functions.
With F. Acquistapace, F. Broglia and J.F. Fernando.

Abstract: We prove that if Hilbert's 17th Problem for global analytic functions has a positive solution for the affine space, then the Pythagoras numbers of the rings (i) of global meromorphic functions and (ii) of meromorphic function germs, are both finite. This is a measure of the difficulty of the problem for analytic functions in the non-compact case. (Click for PDF.)

Keywords: Hilbert's 17th Problem, Pythagoras number, infinite sum of squares, bad set, germs at closed sets.

Mathematical Subject Classification: 14P99, 11E25, 32B10, 32S05.


[04a] On the Hilbert 17th Problem for global analytic functions.
With F. Acquistapace, F. Broglia and J.F. Fernando.

Abstract: We consider Hilbert's 17th Problem for global analytic functions in a modified form that involves infinite sums of squares. This reveals an essential connection between the solution of the problem and the computation of Pythagoras numbers of meromorphic functions. (Click for PDF.)

Keywords: Hilbert's 17th Problem, Pythagoras number, infinite sum of squares, bad set, germs at closed sets.

Mathematical Subject Classification: 14P99, 11E25, 32B10.