| A positive semidefinite polynomial {dollar}finIRlbrack xsb1,...,xsb{lcub}d{rcub}rbrack{dollar} is said to be (m, n) if f is a sum of m squares in {dollar}IR(xsb1,...,xsb{lcub}d{rcub}),{dollar} but no fewer, and f is a sum of n squares in {dollar}IRlbrack xsb1,...,xsb{lcub}d{rcub}rbrack,{dollar} but no fewer. If f is not a sum of polynomial squares, then {dollar}n=infty.{dollar} It is known that {dollar}mle2sp{lcub}n{rcub}.{dollar} Given a particular value of m, what values of n are possible? The case {dollar}mle2{dollar} has been characterized completely. We present a partial answer in the case {dollar}m=3.{dollar} Specifically, we exhibit a family of (3, 4) polynomials and a family of (3, {dollar}infty){dollar} polynomials. Thus, a positive semidefinite polynomial in two variables may be a sum of three rational squares, and yet not be a sum of polynomial squares. This resolves in part a problem posed by Choi, Lam, Reznick, and Rosenberg in J. Algebra 65, p. 254 (1980).; Given a ring R and a subset {dollar}Ssubset R,{dollar} the pythagoras number of S is the number{dollar}{dollar}P(S)=sup{lcub}{lcub}rm length{rcub}(f)vert f {lcub}rm is a sum of squares in{rcub} R{rcub},{dollar}{dollar}where length(f) represents the minimum number of squares required to represent f as a sum of squares.; We follow Choi, Lam, and Reznick in denoting by {dollar}Csb{lcub}n,m{rcub}{dollar} the subset of {dollar}IRlbrack xsb1,...,xsb{lcub}n{rcub}rbrack{dollar} made up of homogeneous polynomials of degree m. In "Sums of Squares of Real Polynomials," Proc. Sym. Pure Math., Amer. Math Soc., 58.2 (1995), 103-126, these authors give upper and lower bound estimates for {dollar}P(Csb{lcub}n,m{rcub}).{dollar} We improve certain of the lower bound estimates both through examples and through geometrical reasoning. |
