On the finiteness of Pythagoras numbers of real meromorphic functions

Abstract

Abstract. -We consider the 17 th 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 17 th 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 meromorphic power series. This measures the difficulty of the 17 th Hilbert problem in the analytic case.

On a global analytic Positivstellensatz

“…Notice that since there is no known bound for the least number of squares needed to represent a sum of squares of meromorphic functions (see [4]), we cannot state similar results to Theorems 1.7, 1.8, 1.9 and Corollary 1.10 for the property PSS. Such a least number of squares refers to the study of the finiteness property (that is, every sum of squares is a finite sum of squares) and the computation of Pythagoras numbers of rings of meromorphic functions (for more details see [4]).…”

“…Indeed, since PSS ∞ holds at Y α for all α, we deduce, by Theorem 1.7, that wPSS holds for {g 1 , ..., g r ; m}. Next, by [4,Theorem 1.5], wehave that any positive semidefinite analytic function f : R n !R whose zero set is Y is a (possibly infinite) sum of squares of meromorphic functions on R n with controlled bad set. Finally, putting all together and clearing denominators we conclude that PSS ∞ holds for {g 1 , ..., g r ; m}, as wanted.…”

“…Thus, our purpose now should be to modify the functions ψ ν and p to obtain an analogous equation to (4), but involving in this case analytic functions whose zero sets are again Y 0 . For that aim, we also introduce the continuous functions η ν on R n given by…”

“…Concerning the difference between arbitrary and controlled bad sets, we recall the following result. For more details about the 17th Hilbert problem for global analytic functions, see [4] and [12].…”

“…Such a least number of squares refers to the study of the finiteness property (that is, every sum of squares is a finite sum of squares) and the computation of Pythagoras numbers of rings of meromorphic functions (for more details see [4]). …”