On Kuroda's proof of Hilbert's fourteenth problem in dimensions three and four

In 1888, Hilbert proved that every nonnegative quartic form $f=f(x,y,z)$ with real coefficients is a sum of three squares of quadratic forms. His proof was ahead of its time and used advanced methods from topology and algebraic geometry. In a 2012 paper we presented a new approach that used only elementary techniques. In this note we add some further simplifications to this proof.

In [1], Theorem 3, the authors proved, in one dimension, a generalization of the Hopf Lemma, and the question arose if it could be extended to higher dimensions. In this paper we present two conjectures as possible extensions, and give a very partial answer. We write this paper to call attention to the problem.

We present some questions and suggestion on the second part of the Hilbert 16th problem

The purpose of this paper is to present a generalization of Forelli's theorem. In particular, we prove an all dimensional version of the two-dimensional theorem of Chirka of 2005.

We prove a generalization of Istvan F\'ary's celebrated theorem to higher dimension.

This paper proposes a totally constructive approach for the proof of Hilbert's theorem on ternary quartic forms. The main contribution is the ladder technique, with which the Hilbert's theorem is proved vividly.

This is the announcement of an alternative approach to the 3-dimensional Poincar\'e Conjecture, different from Perelman's big and spectacular breakthrough. No claim concerning the other parts of the Thurston Geometrization Conjecture, come with our purely 4-dimensional line of argument.

This paper provides a new simple proof of Hesse's theorem in projective geometry for any dimension.

We interpret a counterexample to Hilbert's 14th problem by S. Kuroda geometrically in two ways: As ring of regular functions on a smooth rational quasiprojective variety over any field K of characteristic 0, and, in the special case where K are the real numbers R, as the ring of bounded polynomials on a regular semialgebraic subset of R^3. One motivation for this was to find a regular semialgebraic subset of a real vectorspace, such that the ring of bounded polynomials on it ...

This survey article collects a few of my favorite open problems of Branko Gr\"{u}nbaum.