Introduction Conclusions References

Suppose S is a surface in P, and p1, . . . , pr are general points on S. What is the dimension of the space of sections of OS(e) having singularities of multiplicity mi at pi for all i? We formulate two natural conjectures which would answer this question, and we show they are equivalent. We then prove these conjectures in case all multiplicities are at most 4.

This paper contains a review of available methods for establishing improved interpolation inequalities on the sphere for subcritical exponents. Pushing further these techniques we also establish some new results, clarify the range of applicability of the various existing methods and state several explicit estimates.

In the paper, a natural class of logical calculi is fixed for which we formulate the notion of a ∆–decomposable set of formulas. We demonstrate that the property of uniqueness of signature decompositions holds in those calculi of this class that have the Craig interpolation property. In conclusion, we give a sufficient condition for the ∆–decomposability property to be decidable.