Rudiments of algebraic geometry. The number of points in varieties over finite fields

1976 ◽  
pp. 216-264
Author(s):  
Wolfgang M. Schmidt
Keyword(s):  
Algebraic Geometry ◽  
Finite Fields

scholarly journals Recognizing Graph Theoretic Properties with Polynomial Ideals

10.37236/386 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Jesús A. De Loera ◽  
Christopher J. Hillar ◽  
Peter N. Malkin ◽  
Mohamed Omar
Keyword(s):  
Algebraic Geometry ◽  
Convex Programming ◽  
Linear Algebra ◽  
Finite Fields ◽  
Combinatorial Problems ◽  
Polynomial Method ◽  
Polynomial Equations ◽  
Graph Theoretic ◽  
Polynomial Ideals ◽  
System Of Polynomial Equations

Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect $k$-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Gröbner bases, toric algebra, convex programming, and real algebraic geometry.

scholarly journals Belyi’s Theorems in Positive Characteristic

2020 ◽  
Vol 16 (06) ◽  
pp. 1355-1368
Author(s):  
Nurdagül Anbar ◽  
Seher Tutdere
Keyword(s):  
Algebraic Geometry ◽  
Finite Fields ◽  
Positive Characteristic ◽  
Function Fields ◽  
Constructive Proof ◽  
Characteristic Two

There are two types of Belyi’s Theorems for curves defined over finite fields of characteristic [Formula: see text], namely the Wild and the Tame [Formula: see text]-Belyi Theorems. In this paper, we discuss them in the language of function fields. In particular, we provide a constructive proof for the existence of a pseudo-tame element introduced in [Y. Sugiyama and S. Yasuda, Belyi’s theorem in characteristic two, Compos. Math. 156(2) (2020) 325–339], which leads to a self-contained proof for the Tame [Formula: see text]-Belyi Theorem. Moreover, we provide unified and simple proofs for Belyi’s Theorems unlike the known ones that use technical results from Algebraic Geometry.

scholarly journals On the tensor rank of multiplication in finite extensions of finite fields and related issues in algebraic geometry

2021 ◽  
Vol 76 (1) ◽  
pp. 29-89
Author(s):  
S Ballet ◽  
J Pieltant ◽  
M Rambaud ◽  
H Randriambololona ◽  
R Rolland ◽  
...  
Keyword(s):  
Algebraic Geometry ◽  
Finite Fields ◽  
Tensor Rank

Finite Fields

1996 ◽  
Author(s):  
Rudolf Lidl ◽  
Harald Niederreiter
Keyword(s):  
Finite Fields
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