Rational points on diagonal cubic surfaces Rational points on diagonal cubic surfaces

It is known that there are diagonal cubic surfaces defined over Q which do not contain rational points, although for each prime p they do contain p-adic points. The simplest example is

3X03+4X13+10X23+15X33 = 0.
It is natural to ask what condition additional to solubility in each p-adic field is needed to ensure solubility in Q; and there is strong numerical evidence that this condition is that there is no Brauer-Manin obstruction. However, one does not know how to exploit such a condition.

Provided one assumes the finiteness of the Tate-Safarevic groups of elliptic curves, I shall show that there is a sufficient condition for solubility which is only slightly stronger than the vanishing of the Brauer-Manin obstruction, and which can be expressed in very straightforward terms.

Designing efficient Liapounov functions

Let [(xi)\dot] = fi(x1,¼,xn) for 1 £ i £ n be a system of first order ordinary differential equations in R. A function V(x1,¼,xn) is called a Liapounov function for the system if

W = å
 		V
xi
 fi
satisfies W < 0 whenever V ³ 0. Liapounov's theorem in its simplest form states that if V is a Liapounov function then (subject to some abuse of language in respect of behaviour at infinity) every trajectory eventually enters the set {V £ 0} and never thereafter leaves it.

The construction of efficient Liapounov functions is currently a hit-and-miss affair. This seminar is a contribution towards making the process more systematic.

File translated from TEX by TTH, version 2.25.
On 02 May 2000, 19:48.