The focus of this mathematical research is on affine varieties in the space of several complex variables. Varieties are the locus of points where analytic functions vanish; they have many similarities with surfaces. The object of the work concerns the Lojasiewicz inequality for complex polynomials. The inequality relates the distance of any complex variable (n- dimensional) from the zero locus of an analytic function to the value of the function. In its sharper form, one wants to know the highest power of the distance function which can be compared with the function value, and, instead of a single function, the zero set belongs to that of a finitely generated ideal. If the generators are polynomials, then a recent result gives a power in terms of the product of the polynomial degrees. A better exponent is conjectured depending on whether the number of polynomials exceeds the dimension of the underlying space or not. Work will be done to establish the conjecture. A second line of investigation relates to the development of a Nevanlinna theory in several variables. That is, the study of value distribution theory or the measure of the affinity of complex meromorphic functions for complex numbers. The specific objects of study are maps from one complex domain to another. The components of the maps are polynomials. This work will continue recent investigations into the Jacobian conjecture to the effect that if the Jacobian of such a map is unity, then the map is an automorphism. While this conjecture may not be settled for some time, accomplishments of the past year suggest that substantial insights may be gained with further research using Nevanlinna-type arguments.
