An elementary proof and an extension of Thas' theorem on k-arcs

1989 ◽  
Vol 105 (3) ◽  
pp. 459-462 ◽  
Author(s):  
Hitoshi Kaneta ◽  
Tatsuya Maruta
Keyword(s):  
Finite Field ◽  
Projective Space ◽  
Elementary Proof ◽  
Rational Curve ◽  
Image Position

Let q be the finite field of q elements. Denote by Sr q the projective space of dimension r over q. In Sr,q, where r ≥ 2, a k-arc is defined (see [4]) as a set of k points such that no j + 2 lie in a Sj,q, for j = 1,2,…, r−1. (For a k-arc with k > r, this last condition holds for all j when it holds for j = r−1.) A rational curve Cn of order n in Sr,q, is the set

scholarly journals THE INVARIANTS FIELD OF SOME FINITE PROJECTIVE LINEAR GROUP ACTIONS

2011 ◽  
Vol 85 (1) ◽  
pp. 19-25
Author(s):  
YIN CHEN
Keyword(s):  
Finite Field ◽  
Projective Space ◽  
Vector Space ◽  
Orthogonal Group ◽  
Homogeneous Polynomial ◽  
Classical Group ◽  
Projective Group ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Projective Linear Group

AbstractLet Fq be a finite field with q elements, V an n-dimensional vector space over Fq and 𝒱 the projective space associated to V. Let G≤GLn(Fq) be a classical group and PG be the corresponding projective group. In this note we prove that if Fq (V )G is purely transcendental over Fq with homogeneous polynomial generators, then Fq (𝒱)PG is also purely transcendental over Fq. We compute explicitly the generators of Fq (𝒱)PG when G is the symplectic, unitary or orthogonal group.

scholarly journals ON THE STABILITY OF SYZYGY BUNDLES

2011 ◽  
Vol 22 (04) ◽  
pp. 515-534 ◽  
Author(s):  
IUSTIN COANDĂ
Keyword(s):  
Projective Space ◽  
Complete Intersection ◽  
Vector Bundles ◽  
Elementary Proof ◽  
Base Point ◽  
Vector Spaces ◽  
Similar Argument ◽  
Vanishing Theorem ◽  
The Stability ◽  
Koszul Cohomology

We are concerned with the problem of the stability of the syzygy bundles associated to base-point-free vector spaces of forms of the same degree d on the projective space of dimension n. We deduce directly, from M. Green's vanishing theorem for Koszul cohomology, that any such bundle is stable if its rank is sufficiently high. With a similar argument, we prove the semistability of a certain syzygy bundle on a general complete intersection of hypersurfaces of degree d in the projective space. This answers a question of H. Flenner [Comment. Math. Helv.59 (1984) 635–650]. We then give an elementary proof of H. Brenner's criterion of stability for monomial syzygy bundles, avoiding the use of Klyachko's results on toric vector bundles. We finally prove the existence of stable syzygy bundles defined by monomials of the same degree d, of any possible rank, for n at least 3. This extends the similar result proved, for n = 2, by L. Costa, P. Macias Marques and R. M. Miro-Roig [J. Pure Appl. Algebra214 (2010) 1241–1262]. The extension to the case n at least 3 has been also, independently, obtained by P. Macias Marques in his thesis [arXiv:0909.4646/math.AG (2009)].

Projective Geometries that are Disjoint Unions of Caps

1980 ◽  
Vol 32 (6) ◽  
pp. 1299-1305 ◽  
Author(s):  
Barbu C. Kestenband
Keyword(s):  
Finite Field ◽  
Projective Geometry ◽  
Prime Power ◽  
Disjoint Union ◽  
Hermitian Matrices ◽  
Incidence Relation ◽  
Projective Geometries ◽  
Image Position ◽  
Natural Way ◽  
Square Matrix

We show that any PG(2n, q2) is a disjoint union of (q2n+1 − 1)/ (q − 1) caps, each cap consisting of (q2n+1 + 1)/(q + 1) points. Furthermore, these caps constitute the “large points” of a PG(2n, q), with the incidence relation defined in a natural way.A square matrix H = (hij) over the finite field GF(q2), q a prime power, is said to be Hermitian if hijq = hij for all i, j [1, p. 1161]. In particular, hii ∈ GF(q). If if is Hermitian, so is p(H), where p(x) is any polynomial with coefficients in GF(q).Given a Desarguesian Projective Geometry PG(2n, q2), n > 0, we denote its points by column vectors:All Hermitian matrices in this paper will be 2n + 1 by 2n + 1, n > 0.

On General Polynomials

1967 ◽  
Vol 10 (4) ◽  
pp. 579-583 ◽  
Author(s):  
Kenneth S. Williams
Keyword(s):  
Finite Field ◽  
Number Of Solutions ◽  
Fixed Integer ◽  
Image Position

Let d denote a fixed integer > 1 and let GF(q) denote the finite field of q = pn elements. We consider q fixed ≥ A(d), where A(d) is a (large) constant depending only on d. Let1where each aiεGF(q). Let nr(r = 2, 3, …, d) denote the number of solutions in GF(q) offor which x1, x2, …, xr are all different.

A Specialised Net of Quadrics Having Selfpolar Polyhedra, with Details of the Fivedimensional Example

1981 ◽  
Vol 33 (4) ◽  
pp. 885-892
Author(s):  
W. L. Edge
Keyword(s):  
Projective Space ◽  
Complex Number ◽  
Roots Of Unity ◽  
Parametric Form ◽  
Normal Curve ◽  
Homogeneous Coordinates ◽  
Image Position ◽  
Rational Normal Curve

If x0,x1, … xn are homogeneous coordinates in [n], projective space of n dimensions, the prime (to use the standard name for a hyperplane)osculates, as θ varies, the rational normal curve C whose parametric form is [2, p. 347]Take a set of n + 2 points on C for which θ = ηjζ where ζ is any complex number andso that the ηj, for 0 ≦ j < n + 2, are the (n + 2)th roots of unity. The n + 2 primes osculating C at these points bound an (n + 2)-hedron H which varies with η, and H is polar for all the quadrics(1.1)in the sense that the polar of any vertex, common to n of its n + 2 bounding primes, contains the opposite [n + 2] common to the residual pair.

scholarly journals On the arithmetic of a family of degree - two K3 surfaces

2018 ◽  
Vol 166 (3) ◽  
pp. 523-542 ◽  
Author(s):  
FLORIAN BOUYER ◽  
EDGAR COSTA ◽  
DINO FESTI ◽  
CHRISTOPHER NICHOLLS ◽  
MCKENZIE WEST
Keyword(s):  
Projective Space ◽  
Function Field ◽  
Weighted Projective Space ◽  
Module Structure ◽  
K3 Surface ◽  
Galois Module ◽  
Generic Element ◽  
The Family ◽  
Formula Group ◽  
Image Position

AbstractLet ℙ denote the weighted projective space with weights (1, 1, 1, 3) over the rationals, with coordinates x, y, z and w; let $\mathcal{X}$ be the generic element of the family of surfaces in ℙ given by \begin{equation*} X\colon w^2=x^6+y^6+z^6+tx^2y^2z^2. \end{equation*} The surface $\mathcal{X}$ is a K3 surface over the function field ℚ(t). In this paper, we explicitly compute the geometric Picard lattice of $\mathcal{X}$, together with its Galois module structure, as well as derive more results on the arithmetic of $\mathcal{X}$ and other elements of the family X.

scholarly journals An Elementary Proof of the Theorems of Cauchy and Mayer

1935 ◽  
Vol 4 (3) ◽  
pp. 112-117
Author(s):  
A. J. Macintyre ◽  
R. Wilson
Keyword(s):  
Differential Equation ◽  
Existence Theorem ◽  
Elementary Proof ◽  
Theoretical Discussion ◽  
Practical Advantage ◽  
Total Differential ◽  
Method Of Solution ◽  
The Subject ◽  
Image Position ◽  
Natural Way

Attention has recently been drawn to the obscurity of the usual presentations of Mayer's method of solution of the total differential equationThis method has the practical advantage that only a single integration is required, but its theoretical discussion is usually based on the validity of some other method of solution. Mayer's method gives a result even when the equation (1) is not integrable, but this cannot of course be a solution. An examination of the conditions under which the result is actually an integral of equation (1) leads to a proof of the existence theorem for (1) which is related to Mayer's method of solution in a natural way, and which moreover appears to be novel and of value in the presentation of the subject.

Distinct Values of a Polynomial in Subsets of a Finite Field

1969 ◽  
Vol 21 ◽  
pp. 1483-1488
Author(s):  
Kenneth S. Williams
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Finite Number ◽  
Image Position

If A is a set with only a finite number of elements, we write |A| for the number of elements in A. Let p be a large prime and let m be a positive integer fixed independently of p. We write [pm] for the finite field with pm elements and [pm]′ for [pm] – {0}. We consider in this paper only subsets H of [pm] for which |H| = h satisfies1.1If f(x) ∈ [pm, x] we let N(f; H) denote the number of distinct values of y in H for which at least one of the roots of f(x) = y is in [pm]. We write d(d ≥ 1) for the degree of f and suppose throughout that d is fixed and that p ≧ p0(d), for some prime p0, depending only on d, which is greater than d.

THE REGULAR PART OF A SEMIGROUP OF LINEAR TRANSFORMATIONS WITH RESTRICTED RANGE

2017 ◽  
Vol 103 (3) ◽  
pp. 402-419 ◽  
Author(s):  
WORACHEAD SOMMANEE ◽  
KRITSADA SANGKHANAN
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Dimensional Subspace ◽  
Finite Dimensional Subspace ◽  
Restricted Range ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Finite Dimensional ◽  
Idempotent Rank ◽  
Image Position

Let$V$be a vector space and let$T(V)$denote the semigroup (under composition) of all linear transformations from$V$into$V$. For a fixed subspace$W$of$V$, let$T(V,W)$be the semigroup consisting of all linear transformations from$V$into$W$. In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’,Bull. Aust. Math. Soc.77(3) (2008), 441–453] proved that$$\begin{eqnarray}\displaystyle Q=\{\unicode[STIX]{x1D6FC}\in T(V,W):V\unicode[STIX]{x1D6FC}\subseteq W\unicode[STIX]{x1D6FC}\} & & \displaystyle \nonumber\end{eqnarray}$$is the largest regular subsemigroup of$T(V,W)$and characterized Green’s relations on$T(V,W)$. In this paper, we determine all the maximal regular subsemigroups of$Q$when$W$is a finite-dimensional subspace of$V$over a finite field. Moreover, we compute the rank and idempotent rank of$Q$when$W$is an$n$-dimensional subspace of an$m$-dimensional vector space$V$over a finite field$F$.

scholarly journals A proof of the “Theorem of the Means.”

1943 ◽  
Vol 33 ◽  
pp. 17-18
Author(s):  
C. E. Walsh
Keyword(s):  
Elementary Proof ◽  
Image Position ◽  
Proof By Induction

Numerous proofs have been given of this familiar theorem, which states that if a1, a2, …, an are positive, and not all equal, thenThe following is an elementary proof by induction, which I have not seen used before. It is, of course, not claimed to be novel, and not likely to be so.

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