Mathematical Sciences: Complex Manifolds and Meromorphic Mappings

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This project continues mathematical research on problems related to the theory of functions of several complex variables. It will focus on the issue of separate analyticity of meromorphic functions. Although Hartog's theorem illustrates that separate analyticity implies joint analyticity for functions defined in domains, one cannnot make such broad statements if the separate analyticity is confined to smaller sets. The present work seeks to expand recent results on restricted analyticity of meromorphic mappings of projective algebraic manifolds to mappings into compact Kahler manifolds. A second line of investigation concerns bounds for degrees in solving polynomial equations. If an ideal is generated by polynomials of limited degree, a polynomial in the ideal may be expressed as combinations of products of the generators (with other polynomials). The minimal degree of the multipliers is not known, although sharp results can be obtained if the locus of roots of the generators can be measured. Work will continue in an effort to improve the bounds under less restrictive hypotheses. //