A Linear Counterexample to the Fourteenth Problem of Hilbert in Dimension Eleven

Counterexamples to the Fourteenth Problem of Hilbert

Let K be a field, K[X] = K[X1, . . . , Xn] the polynomial ring in n variables over K for some n ∈ N, and K(X) the field of fractions of K[X]. Assume that L is a subfield of K(X) containing K. Then, the Fourteenth Problem of Hilbert asks whether the Ksubalgebra L ∩K[X] of K[X] is finitely generated. Zariski [17] showed in 1954 that the answer to this problem is affirmative if the transcendence d...

On the postulation of s^d fat points in P^d

In connection with his counterexample to the fourteenth problem of Hilbert, Nagata formulated a conjecture concerning the postulation of r fat points of the same multiplicity in P 2 and proved it when r is a square. Iarrobino formulated a similar conjecture in P d. We prove Iarrobino's conjecture when r is a d-th power. As a corollary, we obtain new counterexamples modeled on those by Nagata.

Hilbert’s fourteenth problem over finite fields, and a conjecture on the cone of curves

Hilbert’s fourteenth problem asks whether the ring of invariants of any representation of a linear algebraic group is finitely generated over the base field. Nagata gave the first counterexample, using a representation of (Ga) 13, where Ga denotes the additive group [22]. In his example, the representation is defined over a field of large transcendence degree over the prime field (of any charac...

Counterexample to Hilbert's fourteenth problem for the 3-dimensional additive group

Strong convergence of variational inequality problem Over the set of common fixed points of a family of demi-contractive mappings

In this paper, by using the viscosity iterative method and the hybrid steepest-descent method, we present a new algorithm for solving the variational inequality problem. The sequence generated by this algorithm is strong convergence to a common element of the set of common zero points of a finite family of inverse strongly monotone operators and the set of common fixed points of a finite family...