I am fairly new to sage, I was studying zeta functions of hypersurfaces over finite fields and I don't know how to compute them in Sage. Are there any packages that do most of the work, or maybe some similar work that I could have a look at to get any ideas?
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$\begingroup$ Smooth projective hypersurfaces? $\endgroup$– Will SawinCommented Sep 15, 2019 at 13:00
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$\begingroup$ Jan Tuitman used to have good code for hypersurfaces, but it never made it into sage for some strange reason. $\endgroup$– Chris WuthrichCommented Sep 15, 2019 at 21:33
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$\begingroup$ @WillSawin smooth projective hypersurfaces are indeed part of the things that I am trying to implement, specifically the zeta function in its rational form. $\endgroup$– Martin OrtizCommented Sep 17, 2019 at 16:43
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1$\begingroup$ @MartinOrtiz The advantage in that case is that you know a priori the degree of the factors in the zeta function (as well as all terms but one), so if your plan is to calculate the coefficients, you only need to do a controlled number of coefficients (in fact, half the dimension of the primitive cohomology group, rounded up) to get the whole rational function. $\endgroup$– Will SawinCommented Sep 17, 2019 at 18:04
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Sage has a method zeta_series() for arbitrary varieties over finite fields, but it will only give you the first few terms of the power series, not the rational-function closed form; the latter is only available for elliptic and hyperelliptic curves if I remember correctly. Perhaps Singular or Macaulay2 (which are more specifically geometry-oriented) might have more relevant functionality.
answered Sep 15, 2019 at 12:53
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$\begingroup$ Searching for documentation, I found this. Relevant section on page 13. It looks like you're referring to zeta_function(), which does return the rational-function closed form. It says it gives the zeta function of a "generic scheme". However the example given returns "NotImplementedError". I imagine the power series from zeta_series() would be useful for finding the rational function if you know its shape (degrees in numerator and denominator, etc.). $\endgroup$ Commented Sep 17, 2019 at 3:57