On Hilbert's construction of positive polynomials

In 1888, Hilbert described how to find real polynomials in more than one variable which take only non-negative values but are not a sum of squares of polynomials. His construction was so restrictive that no explicit examples appeared until the late 1960s. We revisit and generalize Hilbert's construction and present many such polynomials

Some new canonical forms for polynomials

We give some new canonical representations for forms over \cc. For example, a general binary quartic form can be written as the square of a quadratic form plus the fourth power of a linear form. A general cubic form in $(x_1,...,x_n)$ can be written uniquely as a sum of the cubes of linear forms $\ell_{ij}(x_i,...,x_j)$, $1 \le i \le j \le n$. A general ternary quartic form is the sum of the square of a quadratic form and three fourth powers of linear forms. The methods are classical and elementary.Comment: I have spoken about this material under the title "steampunk canonical forms". This is the final revised version which has been accepted by the Pacific Journal of Mathematics. Apart from the usual improvements which come after a thoughtful refereeing, Theorem 1.8 is ne

Laws of inertia in higher degree binary forms

We consider representations of real forms of even degree as a linear combination of powers of real linear forms, counting the number of positive and negative coefficients. We show that the natural generalization of Sylvester's Law of Inertia holds for binary quartics, but fails for binary sextics.Comment: 13 page