Plan of the course <b>Analytic and Cobinatorial Number Theory</b> in summer term 2025/26
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Finite fields and their combimatorial applications
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Counting solutions of diagonal congruences mod p
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An elementary proof of Weil's theorem on numbers of solutions of polynomial congruences mod p
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<a href="AKTC_LN26.pdf">lecture notes</a> (preliminary, updated February 25, 2026)
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Lecture 1, February 16, 2026.
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Finite fields.
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Lecture 2, February 23, 2026.
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Sidon sets
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Lecture 3, March 2, 2026.
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Combinatorial applications of the Chevalley - Warning theorem.
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Lecture 4, March 9, 2026.
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A theorem from Borevich and Shafarevich on the number of solutions of a_1x^{r_1}+...
+a_nx^{r_n} = 0 in F = Z_p (without proof). Beginning of the proof of Theorem 1: 
If N is the # of solutions of P(X, Y)=0 in Z_p, where P(X, Y) is abs. irreducible 
with d = deg(P) such that p > 250d^5, then |N - p|<(2d^5*p)^{1/2}. 
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