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

In 1888, Hilbert proved that every non-negative 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. Up to now, no elementary proof is known. Here we present a completely new approach. Although our proof is not easy, it uses only elementary techniques. As a by-product, it gives information on the number of representations f=p_1^2+p_2^2+p_3^2 ...

Plank problems concern the covering of convex bodies by planks in Euclidean space and are related to famous open problems in convex geometry. In this survey, we introduce plank problems and present surprising applications of plank theorems in various areas of mathematics.

From any configuration of finitely many points in Euclidean three-space, Atiyah constructed a determinant and conjectured that it was always non-zero. Atiyah and Sutcliffe (hep-th/0105179) amass a great deal of evidence it its favour. In this article we prove the conjecture for the case of four points.

The main results of the paper are Proposition 3 and 4 which provide an effective way to construct minimal hypersurfaces in a Euclidean space. We demonstrate our technique by several new examples. This note is English translation of an earlier draft version written (in Russian) in September 1999. The final version of the paper will be published somewhere else.

Admin note: withdrawn by arXiv admin because of the use of a pseudonym, in violation of arXiv policy.

This survey presents recent Helly-type geometric theorems published since the appearance of the last comprehensive survey, more than ten years ago. We discuss how such theorems continue to be influential in computational geometry and in optimization.

We give a new proof of a theorem of Montejano and Karasev regarding $k$-dimensional transversals to small families of convex sets. While their proof uses technical algebraic and topological tools, our proof is a simple application of the Borsuk-Ulam theorem. Additionally, in certain cases our result is stronger than the Montejano-Karasev theorem.

A compendium of thirty previously published open problems in computational geometry is presented.

In [4], Kuroda generalized Roberts' counterexample \cite{Roberts} to the fourteenth problem of Hilbert. The counterexample is given as the kernel of a locally nilpotent derivation on a polynomial ring. We replace his construction of the invariant elements by a more straightforward construction and give a more precise form of invariant elements.

We study the minimal genus problem for some smooth four-manifolds.