Detecting Rational Points on Hypersurfaces over Finite Fields

2008 ◽  
Author(s):  
Swastik Kopparty ◽  
Sergey Yekhanin
Keyword(s):  
Finite Fields ◽  
Rational Points

scholarly journals Construction of elliptic curves with cyclic groups over prime fields

2006 ◽  
Vol 73 (2) ◽  
pp. 245-254 ◽  
Author(s):  
Naoya Nakazawa
Keyword(s):  
Elliptic Curves ◽  
Finite Fields ◽  
Function Field ◽  
Modular Group ◽  
Modular Function ◽  
Invariant Function ◽  
Rational Points ◽  
Modular Invariant ◽  
Cyclic Groups ◽  
Prime Fields

The purpose of this article is to construct families of elliptic curves E over finite fields F so that the groups of F-rational points of E are cyclic, by using a representation of the modular invariant function by a generator of a modular function field associated with the modular group Γ0(N), where N = 5, 7 or 13.

On the curve Yn = Xℓ(Xm + 1) over finite fields

2019 ◽  
Vol 19 (2) ◽  
pp. 263-268 ◽  
Author(s):  
Saeed Tafazolian ◽  
Fernando Torres
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Plane Curve ◽  
Rational Points ◽  
Weil Bound

Abstract Let 𝓧 be the nonsingular model of a plane curve of type yn = f(x) over the finite field F of order q2, where f(x) is a separable polynomial of degree coprime to n. If the number of F-rational points of 𝓧 attains the Hasse–Weil bound, then the condition that n divides q+1 is equivalent to the solubility of f(x) in F; see [20]. In this paper, we investigate this condition for f(x) = xℓ(xm+1).

scholarly journals Congruences for rational points on varieties over finite fields

2005 ◽  
Vol 333 (4) ◽  
pp. 797-809 ◽  
Author(s):  
N. Fakhruddin ◽  
C. S. Rajan
Keyword(s):  
Finite Fields ◽  
Rational Points

Rational points on Fermat curves over finite fields

2017 ◽  
Vol 16 (03) ◽  
pp. 1750046
Author(s):  
Wei Cao ◽  
Shanmeng Han ◽  
Ruyun Wang
Keyword(s):  
Finite Fields ◽  
Rational Point ◽  
Rational Points ◽  
Fermat Curve ◽  
Fermat Curves ◽  
Curves Over Finite Fields

Let [Formula: see text] be the [Formula: see text]-rational point on the Fermat curve [Formula: see text] with [Formula: see text]. It has recently been proved that if [Formula: see text] then each [Formula: see text] is a cube in [Formula: see text]. It is natural to wonder whether there is a generalization to [Formula: see text]. In this paper, we show that the result cannot be extended to [Formula: see text] in general and conjecture that each [Formula: see text] is a cube in [Formula: see text] if and only if [Formula: see text].

A SPECIAL DEGREE REDUCTION OF POLYNOMIALS OVER FINITE FIELDS WITH APPLICATIONS

2011 ◽  
Vol 07 (04) ◽  
pp. 1093-1102 ◽  
Author(s):  
WEI CAO
Keyword(s):  
Finite Field ◽  
Explicit Formula ◽  
Finite Fields ◽  
Rational Points ◽  
Degree Reduction ◽  
Affine Hypersurface ◽  
Special Degree ◽  
Low Degree ◽  
Polynomials Over Finite Fields ◽  
Degree Matrix

Let f be a polynomial in n variables over the finite field 𝔽q and Nq(f) denote the number of 𝔽q-rational points on the affine hypersurface f = 0 in 𝔸n(𝔽q). A φ-reduction of f is defined to be a transformation σ : 𝔽q[x1, …, xn] → 𝔽q[x1, …, xn] such that Nq(f) = Nq(σ(f)) and deg f ≥ deg σ(f). In this paper, we investigate φ-reduction by using the degree matrix which is formed by the exponents of the variables of f. With φ-reduction, we may improve various estimates on Nq(f) and utilize the known results for polynomials with low degree. Furthermore, it can be used to find the explicit formula for Nq(f).

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