Prime Subfield Research Articles

The generating rank of a polar Grassmannian

Abstract In this paper we compute the generating rank of k-polar Grassmannians defined over commutative division rings. Among the new results, we compute the generating rank of k-Grassmannians arising from Hermitian forms of Witt index n defined over vector spaces of dimension N > 2n. We also study generating sets for the 2-Grassmannians arising from quadratic forms of Witt index n defined over V(N, 𝔽 q ) for q = 4, 8, 9 and 2n ≤ N ≤ 2n + 2. We prove that for N > 6 and anisotropic defect (polar corank) d ≠ 2 they can be generated over the prime subfield, thus determining their generating rank.

Generalised linear orthomorphic permutations

The concept of generalised orthomorphic permutations over the Galois field Fnq is proposed on the basis of the original concept with regard to orthomorphic permutations over the Galois field Fn2 in this scientific research paper. At the same time, the enumeration formula of the total generalised linear orthomorphic permutations over the Galois field Fnq is given. It is worth noting that the Galois field Fnq possesses the prime subfield Fp with an arbitrary prime number as the characteristic. The local creating algorithm is obtained and generates the partial generalised linear orthomorphic permutations over the Galois general field Fnq. Consequently the existing and innovative enumeration formula involved linear orthomorphic permutations over a Galois field with characteristic two happens to be a special case to contain in our novel fruits over the general field. In the field of information security, the generalised linear orthomorphic permutations have been thoroughly discussed and generated far and wide for the design of the p-permutation in the diffusion layer of iterative round function involving an unbreakable cryptosystem. Finally we can obtain some generalised linear orthomorphic permutations with the greatest branch number in this research paper.

Stable range one for rings with central units

The purpose of this paper is to give a partial positive answer to a question raised by Khurana et al. as to whether a ring R with stable range one and central units is commutative. We show that this is the case under any of the following additional conditions: R is semiprime or R is one-sided Noetherian or R has unit-stable range 1 or R has classical Krull dimension 0 or R is an algebra over a field K such that K is uncountable and R has only countably many primitive ideals or R is affine and either K has characteristic 0 or has infinite transcendental degree over its prime subfield or is algebraically closed. However, the general question remains open.

Directed evaluation

Let K be a fixed effective field. The most straightforward approach to compute with an element in the algebraic closure of K is to compute modulo its minimal polynomial. The determination of a minimal polynomial from an arbitrary annihilator requires an algorithm for polynomial factorization over K. Unfortunately, such algorithms do not exist over generic effective fields. They do exist over fields that are explicitly generated over their prime sub-field, but they are often expensive. The dynamic evaluation paradigm, introduced by Duval and collaborators in the eighties, offers an alternative algorithmic solution for computations in the algebraic closure of K. This approach does not require an algorithm for polynomial factorization, but it still suffers from a non-trivial overhead due to suboptimal recomputations. For the first time, we design another paradigm, called directed evaluation, which combines the conceptual advantages of dynamic evaluation with a good worst case complexity bound.

On the Representation of Fields as Finite Sums of Proper Subfields

We study which fields F can be represented as finite sums of proper subfields. We prove that for any n ge 2 every field F of infinite transcendence degree over its prime subfield can be represented as an unshortenable sum of n subfields, and every rational function field F = K(x_1, ldots , x_n) can be represented as an unshortenable sum of n + 1 subfields. We also show that no subfield of the algebraic closure of a finite field is a finite sum of proper subfields, and no finite extension of the field {mathbb {Q}} of rationals can be decomposed into a sum of two proper subfields.

On Mutual Definability of Operations on Fields

We study the possibilities of defining some operations on fields via the remaining operations. In particular, we prove that multiplication on an arbitrary field can be defined via addition if and only if the field is a finite extension of its prime subfield. We give a sufficient condition for the nondefinability of addition via multiplication and demonstrate that multiplication and addition on the reals and complexes cannot be mutually defined by means of the relations with parameters which are preserved under automorphisms. We also describe the mutual definability of addition, multiplication, and exponentiation via the remaining two operations.

Double sums of Kloosterman sums in finite fields

We bound double sums of Kloosterman sums over a finite field Fq, with one or both parameters ranging over an affine space over its prime subfield Fp⊆Fq. These are finite fields analogues of a series of recent results by various authors in finite fields and residue rings. Our results are based on recent advances in additive combinatorics in arbitrary finite field.

Definable relations in finite dimensional subspace lattices with involution. Part II: Quantifier-free and homogeneous descriptions

For finite dimensional hermitean inner product spaces V, over \(*\)-fields F, and in the presence of orthogonal bases providing form elements in the prime subfield of F, we show that quantifier-free definable relations in the subspace lattice \(\mathsf{L}(V)\), endowed with the involution induced by orthogonality, admit quantifier-free descriptions within F, also in terms of Grassmann–Plucker coordinates. In the latter setting, homogeneous descriptions are obtained if one allows quantification type \(\Sigma _1\). In absence of involution, these results remain valid.

Algebras of linear growth and the dynamical Mordell–Lang conjecture

Ufnarovski remarked in 1990 that it is unknown whether any finitely presented associative algebra of linear growth is automaton, that is, whether the set of normal words in the algebra form a regular language. If the algebra is graded, then the rationality of the Hilbert series of the algebra follows from the affirmative answer to Ufnarovski's question. Assuming that the ground field has a positive characteristic, we show that the answer to Ufnarovskii's question is positive for graded algebras if and only if the basic field is an algebraic extension of its prime subfield. Moreover, in the “only if” part we show that there exists a finitely presented graded algebra of linear growth with irrational Hilbert series. In addition, over an arbitrary infinite basic field, the set of Hilbert series of the quadratic algebras of linear growth with 5 generators is infinite.Our approach is based on a connection with the dynamical Mordell–Lang conjecture. This conjecture describes the intersection of an orbit of an algebraic variety endomorphism with a subvariety. We show that the positive answer to Ufnarovski's question implies some known cases of the dynamical Mordell–Lang conjecture. In particular, the positive answer for a class of algebras is equivalent to the Skolem–Mahler–Lech theorem which says that the set of the zero elements of any linear recurrent sequence over a zero characteristic field is the finite union of several arithmetic progressions. In particular, the counter-examples to this theorem in the positive characteristic case give examples of algebras with irrational Hilbert series.

Finiteness of Brauer groups of K3 surfaces in characteristic 2

For a [Formula: see text] surface over a field of characteristic [Formula: see text] which is finitely generated over its prime subfield, we prove that the cokernel of the natural map from the Brauer group of the base field to that of the [Formula: see text] surface is finite modulo the [Formula: see text]-primary torsion subgroup. In characteristic different from [Formula: see text], such results were previously proved by Skorobogatov and Zarhin. We basically follow their methods with an extra care in the case of superspecial [Formula: see text] surfaces using the recent results of Kim and Madapusi Pera on the Kuga-Satake construction and the Tate conjecture for [Formula: see text] surfaces in characteristic [Formula: see text].

Some results on finite fields

In the study of perfect sequences, use of trace functions in finite fields has been a common technique, dating back to Singer’s classical examples of msequences. Many variations of Singer’s theme have been the topic of research by many mathematicians during the last few decades. In all these investigations, expressing the underlying function as a polynomial whose coefficients are from the prime subfield has been a main problem. Explicit answers require the use of Stickelberger’s congruence of Gauss sums. Using elementary methods, we provide a simple result along these lines. The following is well-known: Let f : F p ? → F p be any function. Then f ( x )=∑ k=0 p−2 α k x ∈ Fp d[x ] We prove a finer version of the above result: Let p be an odd prime. Let f ( x )∈ Fp d [x ] . Let deg f (x )=r . Suppose r< p −1∧f (α )∈F p for each α∈F p ? . Then f ( x )∈ Fp [ x ] .

La conjecture de Tate entière pour les cubiques de dimension quatre

AbstractWe prove the integral Tate conjecture for cycles of codimension$2$on smooth cubic fourfolds over an algebraic closure of a field finitely generated over its prime subfield and of characteristic different from$2$or$3$. The proof relies on the Tate conjecture with rational coefficients, proved in that setting by the first author, and on an argument of Voisin coming from complex geometry.

The Cauchy-Schwarz inequality in Cayley graph and tournament structures on finite fields

The Cayley graph construction provides a natural grid structure on a finite vector space over a field of prime or prime square cardinality, where the characteristic is congruent to 3 modulo 4, in addition to the quadratic residue tournament structure on the prime subfield. Distance from the null vector in the grid graph defines a Manhattan norm. The Hermitian inner product on these spaces over finite fields behaves in some respects similarly to the real and complex case. An analogue of the Cauchy-Schwarz inequality is valid with respect to the Manhattan norm. With respect to the non-transitive order provided by the quadratic residue tournament, an analogue of the Cauchy-Schwarz inequality holds in arbitrarily large neighborhoods of the null vector, when the characteristic is an appropriate large prime. 2010 Mathematics Subject Classification: 05C12; 05C20; 05C25; 06F99; 11T99

A construction of rational elliptic surfaces with the non-surjective boundary map on K 2

We present some rational elliptic surfaces over a certain field such that certain open subschemes do not satisfy the surjectivity of the boundary map on K 2 arising from the localization sequence. We consider two cases that the base field is transcendental over its prime subfield and that the base field is characteristic zero.

Simultaneous embeddings of finite dimensional division algebras

A celebrated theorem of P.M.Cohn says that for any two division rings (not necessarily finite dimensional) over a field F, their amalgamated product over F is a domain which can be embedded in a division ring. Note that even with the two initial division rings begin finite dimensional over their centers, the resulting division ring is never finite dimensional over its center. Perhaps this led Lance Small to ask the following question. Assume $F_1$ and $F_2$ are fields with the same characteristic. Small asked whether any two division algebras $D_1/F_1$ and $D_2/F_2$ can be embedded in some third division algebra $E/F$. We start with a surprisingly straightforward counterexample, but then show that a positive solution exists for division algebras finitely generated over a common subfield which is either algebraically closed or the prime subfield.

The local structure of groups of triangular automorphisms of relatively free algebras

Let K be an arbitrary field and C n a relatively free algebra of rank n. In particular, as C n we may treat a polynomial algebra P n , a free associative algebra A n , or an absolutely free algebra F n . For the algebras C n = P n , A n , F n , it is proved that every finitely generated subgroup G of a group TC n of triangular automorphisms admits a faithful matrix representation over a field K; hence it is residually finite by Mal’tsev’s theorem. For any algebra C n , the triangular automorphism group TC n is locally soluble, while the unitriangular automorphism group UC n is locally nilpotent. Consequently, UC n is local (linear and residually finite). Also it is stated that the width of the commutator subgroup of a finitely generated subgroup G of UC n can be arbitrarily large with increasing n or transcendence degree of a field K over the prime subfield.

A finiteness condition for algebras over coordinate rings of varieties

Let A be a finitely presented k[X]-algebra, where k[X] is the algebra of regular functions on a variety X defined over an algebraically closed field k. The following problem arises in the study of degenerations of algebras (Hajduk and Kasjan, in press [7]). Assume that the specialization of A at x has finite dimension over k for every x from a dense subset of X. Is there an open dense subset U of X such that the localization of A with respect to U is a finitely generated k[U]-module? We prove that this is the case if k has infinite transcendence degree over its prime subfield. We provide some applications to the concept of generalized CB-degenerations.

Locally Finite Conditions on Maximal Subgroups of GLn(D)

Given a division ring D with center F, the structure of maximal subgroups M of GLn(D) is investigated. Suppose D ≠ F or n &gt; 1. It is shown that if M/(M ∩ F*) is locally finite, then char F=p &gt; 0 and either n=1, [D:F]=p2 and M ∪ {0} is a maximal subfield of D, or D=F, n=p, and M ∪ {0} is a maximal subfield of Mp(F), or D=F and F is locally finite. It is also proved that the same conclusion holds if M/(M ∩ F*) is torsion and D is of finite dimension over F. Furthermore, it is shown that if the r-th derived group M(r) of M is locally finite, then either M(r) is abelian or F is algebraic over its prime subfield.

On the Bringer–Chabanne EPIR protocol for polynomial evaluation

Extended private information retrieval (EPIR) was defined by \cite{BCPT07} at CANS'07 and generalized by \cite{BC09} at AFRICACRYPT'09. In the generalized setting, EPIR allows a user to evaluate a function on a database block such that the database can learn neither which function has been evaluated nor on which block the function has been evaluated and the user learns no more information on the database blocks except for the expected result. An EPIR protocol for evaluating polynomials over a finite field $L$ was proposed by Bringer and Chabanne in \cite{BC09}. We show that the protocol does not satisfy the correctness requirement as they have claimed. In particular, we show that it does not give the user the expected result with large probability if one of the coefficients of the polynomial to be evaluated is primitive in $L$ and the others belong to the prime subfield of $L$.

Generalized incidence theorems, homogeneous forms and sum–product estimates in finite fields

In recent years, sum–product estimates in Euclidean space and finite fields have received great attention. They can often be interpreted in terms of Erdős type incidence problems involving the distribution of distances, dot products, areas, and so on, which have been studied quite extensively by way of combinatorial and Fourier analytic techniques. We use both kinds of techniques to obtain sharp or near-sharp results on the distribution of volumes (as examples of d -linear homogeneous forms) determined by sufficiently large subsets of vector spaces over finite fields and the associated arithmetic expressions. Arithmetic–combinatorial techniques turn out to be optimal for dimension d ≥ 4 to this end, while for d = 3 they have failed to provide us with a result that follows from the analysis of exponential sums. To obtain the latter result we prove a relatively straightforward function version of an incidence results for points and planes previously established in [D. Hart, A. Iosevich, Sums and products in finite fields: An integral geometric viewpoint, in: Radon Transforms, Geometry, and Wavelets, Contemp. Math. 464 (2008); D. Hart, A. Iosevich, D. Koh, M. Rudnev, Averages over hyperplanes, sum–product theory in vector spaces over finite fields and the Erdős–Falconer distance conjecture, arXiv:math/0711.4427, preprint 2007]. More specifically, we prove that if E = A × ⋯ × A is a product set in F q d , d ≥ 4 , the d -dimensional vector space over a finite field F q , such that the size | E | of E exceeds q d 2 (i.e. the size of the generating set A exceeds q ) then the set of volumes of d -dimensional parallelepipeds determined by E covers F q . This result is sharp as can be seen by taking A = F p , a prime sub-field of its quadratic extension F q , with q = p 2 . For in three dimensions, however, we are able to establish the same result only if | E | ≳ q 15 8 (i.e., | A | ≥ C q 5 8 , for some C ; in fact, the q 15 8 bound can be justified for a slightly wider class of “Cartesian product-like” sets), and this uses Fourier methods. Yet we do prove a weaker near-optimal result in three dimensions: that the set of volumes generated by a product set E = A × A × A covers a positive proportion of F q if | E | > q 3 2 (so | A | > q ). Besides, without any assumptions on the structure of E , we show that in three dimensions the set of volumes covers a positive proportion of F q if | E | ≥ C q 2 , which is again sharp up to the constant C , as taking E to be a 2-plane through the origin shows.