Sums of even powers of k-regulous functions

摘要

© 2022 Royal Dutch Mathematical Society (KWG)We provide an example of a nonnegative k-regulous function on Rn for k≥1 and n≥2 which cannot be written as a sum of squares of k-regulous functions. We then obtain lower bounds for Pythagoras numbers p2d(Rk(Rn)) of k-regulous functions on Rn for k≥1 and n≥2. We also prove that the second Pythagoras number of the ring of 0-regulous functions R0(X) on an irreducible 0-regulous affine variety X is finite and bounded from above by 2dimX.