scholarly journals On a number of rational points on a plane curve of low degree

2017 ◽  
Vol 340 (6) ◽  
pp. 1327-1334
Author(s):  
Eun Ju Cheon ◽  
Masaaki Homma ◽  
Seon Jeong Kim ◽  
Namyong Lee
Keyword(s):  
Plane Curve ◽  
Rational Points ◽  
Low Degree

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 Components of the Hilbert scheme of space curves on low-degree smooth surfaces

2015 ◽  
Vol 26 (02) ◽  
pp. 1550017 ◽  
Author(s):  
Jan O. Kleppe ◽  
John C. Ottem
Keyword(s):  
Hilbert Scheme ◽  
Smooth Surface ◽  
Plane Curve ◽  
Picard Number ◽  
Connected Space ◽  
Space Curves ◽  
Cubic Surfaces ◽  
Smooth Component ◽  
Low Degree ◽  
Smooth Plane

We study maximal families W of the Hilbert scheme, H(d, g)sc, of smooth connected space curves whose general curve C lies on a smooth surface S of degree s. We give conditions on C under which W is a generically smooth component of H(d, g)sc and we determine dim W. If s = 4 and W is an irreducible component of H(d, g)sc, then the Picard number of S is at most 2 and we explicitly describe, also for s ≥ 5, non-reduced and generically smooth components in the case Pic (S) is generated by the classes of a line and a smooth plane curve of degree s - 1. For curves on smooth cubic surfaces the first author finds new classes of non-reduced components of H(d, g)sc, thus making progress in proving a conjecture for such families.

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).

A K3 Surface Associated With Certain Integral Matrices Having Integral Eigenvalues

2006 ◽  
Vol 49 (4) ◽  
pp. 560-577 ◽  
Author(s):  
Ronald van Luijk
Keyword(s):  
Rational Points ◽  
K3 Surface ◽  
Integral Matrices ◽  
Low Degree ◽  
Symmetric Integral

AbstractIn this article we will show that there are infinitely many symmetric, integral 3× 3 matrices, with zeros on the diagonal, whose eigenvalues are all integral. We will do this by proving that the rational points on a certain non-Kummer, singular K3 surface are dense. We will also compute the entire Néron–Severi group of this surface and find all low degree curves on it.

Assessment of Damage to Human Platelets after Aggregation and Other Injuries by Microscopic Observation and Estimation of Serotonin Uptake

1970 ◽  
Vol 23 (02) ◽  
pp. 261-275 ◽  
Author(s):  
G Zbinden ◽  
J. N Mehrishi ◽  
S Tomlin
Keyword(s):  
Platelet Aggregation ◽  
Electrophoretic Mobility ◽  
Human Platelets ◽  
Freezing And Thawing ◽  
Low Degree ◽  
Microscopic Evaluation ◽  
Negative Surface ◽  
Negative Surface Charge ◽  
Charged Macromolecules ◽  
5 Ht Uptake

SummaryThe severity of platelet damage induced by hyper- and hypotonic NaCl solutions and freezing and thawing was assessed by microscopic evaluation and measuring inhibition of 5-HT uptake. The same techniques were used to quantitate the effects of aggregating agents. The positively charged macromolecules PS, Poly-L und Poly-O reduced the net negative surface charge as determined by microelectrophoresis, caused platelet aggregation and inhibited 5-HT uptake. The damaging effects of Poly-L and Poly-O were more severe and more closely related to concentration than that of PS. The negatively charged macromolecules Poly-IC and NaPS increased the anodic electrophoretic mobility. Poly-IC and heparin caused a low degree of platelet clumping and no inhibition of 5-HT uptake. NaPS produced severe platelet damage with extensive clumping and complete inhibition of 5-HT uptake. Na laurate had the same effect, but did not alter electrophoretic mobility. ADP caused concentration-dependent platelet aggregation and inhibition of 5-HT uptake. The effects of ADP and NaPS were compared in agitated and non-agitated platelet samples containing identical concentrations of the 2 compounds. Agitation was found to increase the degree of platelet clumping and to reduce 5-HT uptake.

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