Positive polynomials and hyperdeterminants
Let \(F\) be a homogeneous real polynomial of even degree in any number of variables. Here the author considers the problem of giving explicit conditions on the coefficients so that \(F\) is positive definite or positive semi-definite. He gives a necessary condition for positivity, and a sufficient condition for non-negativity, in terms of positivity or semipositivity of a one variable characteristic polynomial. He translates the sufficient condition in terms of Hankel matrices. Conditions are known for degree \(2\) (quadratic forms), and for two variables (binary form). He gives a clever reduction of the general case to these cases. Inside the vector space of all real homogeneous polynomials with fixed degree and number of variables there are the cone \(\Pi\) of positive polynomials and the discriminant hypersurface \(\Delta\). Part of his work is the study of their mutual geometry (connectedness of \(\Pi \backslash \Delta\)).
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