Hodge structure
The existence of such polynomials would follow from the existence of an analogue of Hodge structure in the cohomologies of a general (singular and non-complete) algebraic variety. The novel feature is that the "n"th cohomology of a general variety looks as if it contained pieces of different weights. This led Alexander Grothendieck to his conjectural theory of motives and motivated a search for an extension of Hodge theory, which culminated in the work of Pierre Deligne. He introduced the notion of a mixed Hodge structure, developed techniques for working with them, gave their construction (based on Heisuke Hironaka's resolution of singularities) and related them to the weights on l-adic cohomology, proving the last part of the Weil conjectures.