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- Rationality of the zeta-function over a finite field
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Postgraduate research
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- Information about research theses
- Past research students
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- Entry requirements
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Abstract:
Suppose that we have a multivariate polynomial f defined over a finite field with q elements. We can count the number of roots of f over finite field extensions containing q2 elements, q3 elements and onwards. Denote by Ns the number of roots of f in the field with qs elements. The zeta-function is a power series which expresses Ns in terms of its coefficients.
Weil conjectured interesting properties about the zeta-function which have since been proven. In particular, Dwork showed that the zeta-function is a rational function, the quotient of two polynomials, using p-adic analysis. This tells us that we need only to find the values Ns up to some bound for s, in order to calculate the zeta-function. From this we can find Ns for any s.
We will discuss the ideas behind Dworks proof and the role that p-adic analysis plays in his argument.