Local zeta function: Difference between revisions

Revision as of 23:42, 21 March 2024

Revision as of 21:04, 22 March 2024

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}}</ref>

Making the variable transformation {{math|''ut''&nbsp;{{=}}&nbsp;''q''^−''s'',}} gives

:<math>

\mathit{Z} (V,ut) = \exp

\left( \sum_{k=1}^{\infty} N_k \frac{ut^k}{k} \right)

</math>

as the [[formal power series]] in the variable <math>u</math>.

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</math>

:<math>

(2)\ \ \frac{d}{dudt} \log \mathit{Z} (V,ut) = \sum_{k=1}^{\infty} N_k ut^{k-1}\ .</math>

In other words, the local zeta function {{math|''Z''(''V'',&nbsp;''ut'')}} with coefficients in the [[finite field]] {{math|'''F'''_''q''}} is defined as a function whose [[logarithmic derivative]] generates the number {{math|''N''_''k''}} of solutions of the equation defining {{mvar|V}} in the degree {{mvar|k}} extension {{math|'''F'''_''q''^''k''.}}

<!--In [[number theory]], a '''local zeta function'''

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:<math>[ F_k : F ] = k \,</math>,

for ''k'' = 1, 2, ... . When ''F'' hasis the unique field with ''q'' elements, ''F_k'' hasis the unique field with <math>q^k</math> elements. Given a set of polynomial equations &mdash; or an [[algebraic variety]] ''V'' &mdash; defined over ''F'', we can count the number

:<math>N_k \,</math>