Lower bounds for Pythagoras numbers of function fields

We show that the transcendence degree of a real function field over an arbitrary real base field is a strict lower bound for its Pythagoras number and a weak lower bound for all its higher Pythagoras numbers.

[1]  Tsit Yuen Lam,et al.  Introduction To Quadratic Forms Over Fields , 2004 .

[2]  Qing Liu,et al.  Algebraic Geometry and Arithmetic Curves , 2002 .

[3]  D. Eisenbud.  Commutative Algebra: with a View Toward Algebraic Geometry , 1995 .

[4]  J. Schmid.  Sums of fourth powers of real algebraic functions , 1994 .

[5]  W. Kucharz.  Generating ideals in real holomorphy rings , 1991 .

[6]  R. Parimala,et al.  LEVELS OF NON-REAL FUNCTION FIELDS OF REAL RATIONAL SURFACES , 1991 .

[7]  L. Bröcker,et al.  Valuations of function fields form the geometrical point of view. , 1986 .

[8]  T. Willmore.  Algebraic Geometry , 1973, Nature.

[9]  J. Cassels,et al.  On Sums of Squares and on Elliptic Curves over Function Fields , 1971 .

[10]  A. Pfister.  Zur Darstellung definiter Funktionen als Summe von Quadraten , 1967 .

[11]  David Grimm.  A NOTE ON THE PYTHAGORAS NUMBER OF REAL FUNCTION FIELDS , 2012 .

[12]  Ludwig Bröcker,et al.  Constructible sets in real geometry , 1996 .

[13]  B. Reznick,et al.  Sums of 2m-th powers of rational functions in one variable over real closed fields , 1996 .

[14]  A. Pfister.  Quadratic forms with applications to algebraic geometry and topology , 1995 .

[15]  Marie-Françoise Roy,et al.  Real algebraic geometry , 1992 .

[16]  W. Kucharz.  Invertible ideals in real holomorphy rings. , 1989 .

[17]  J. Jouanolou,et al.  Théorèmes de Bertini et applications , 1983 .

[18]  E. Becker.  The real holomorphy ring and sums of 2n-th powers , 1982 .

[19]  E. Becker.  Valuations and real places in the theory of formally real fields , 1982 .