Planar polynomials for commutative semifields with specified nuclei

We consider the implications of the equivalence of commutative semifields of odd order and planar Dembowski-Ostrom polynomials. This equivalence was outlined recently by Coulter and Henderson. In particular, following a more general statement concerning semifields we identify a form of planar Dembowski-Ostrom polynomial which must define a commutative semifield with the nuclei specified. Since any strong isotopy class of commutative semifields must contain at least one example of a commutative semifield described by such a planar polynomial, to classify commutative semifields it is enough to classify planar Dembowski-Ostrom polynomials of this form and determine when they describe non-isotopic commutative semifields. We prove several results along these lines. We end by introducing a new commutative semifield of order 3⁸ with left nucleus of order 3 and middle nucleus of order 3².     

A geometric characterisation of Desarguesian spreads

We construct and describe the basic properties of a family of semifields in characteristic 2. The construction relies on the properties of projective polynomials over finite fields. We start by associating non-associative products to each such polynomial. The resulting presemifields form the degenerate case of our family. They are isotopic to the Knuth semifields which are quadratic over left and right nuclei. The non-degenerate members of our family display a very different behavior. Their left and right nuclei agree with the center, the middle nucleus is quadratic over the center. None of those semifields is isotopic or Knuth equivalent to a commutative semifield. As a by-product we obtain the complete taxonomy of the characteristic 2 semifields which are quadratic over the middle nucleus, bi-quadratic over the left and right nuclei and not isotopic to twisted fields. This includes determining when two such semifields are isotopic and the order of the autotopism group.     

Semifield flat flocks

Using ‘fusion’ methods on finite semifields, a variety of partitions (flocks) of Segre varieties by caps are obtained. The partitions arise from semifield planes and are thus called “semifield flat flocks”. Furthermore, the finite transitive semifield flat flocks are completely determined.     

Fractional dimensions in semifields of odd order

A finite semifield D is considered a fractional dimensional semifield if it contains a subsemifield E such that λ := log_|E||D| is not an integer. We develop spread-theoretic tools to determine when finite planes admit coordinatization by fractional semifields, and to find such semifields when they exist. We use our results to show that such semifields exist for prime powers 3 ⁿ whenever n is an odd integer divisible by 5 or 7.     

The six semifield planes associated with a semifield flock

In 1965 Knuth (J. Algebra 2 (1965) 182) noticed that a finite semifield was determined by a 3-cube array (a_ijk) and that any permutation of the indices would give another semifield. In this article we explain the geometrical significance of these permutations. It is known that a pair of functions (f,g) where f and g are functions from GF(q) to GF(q) with the property that f and g are linear over some subfield and g(x)²+4xf(x) is a non-square for all x∈GF(q)∗, q odd, give rise to certain semifields, one of which is commutative of rank 2 over its middle nucleus, one of which arises from a semifield flock of the quadratic cone, and another that comes from a translation ovoid of Q(4,q). We show that there are in fact six non-isotopic semifields that can be constructed from such a pair of functions, which will give rise to six non-isomorphic semifield planes, unless (f,g) are of linear type or of Dickson-Kantor-Knuth type. These six semifields fall into two sets of three semifields related by Knuth operations.     

The principal kernels of semifields of continuous positive functions

This work is devoted to the study of kernel lattices of semifields of continuous positive functions defined on some topological space. It is established that they are lattices with a pseudo-complement. New characterizations of the following properties of topological spaces are obtained in terms of kernels, predominantly principal kernels, and semifields of continuous functions: to be an F-space, to be a P-space, basic and extremal disconnectedness, pseudo-compactness, and finiteness.     

Albert's Construction for Semifields of Even Order

Albert's construction for commutative semifields of order 2 ⁿ , n odd, is presented. It avoids the construction of a presemifield and, in the case that n is prime, allows us to determine automorphism groups and the isomorphism classes. If n is a prime greater than three, the semifields are strictly not associative. These semifields are new for all n greater than three, differing from the binary semifields in that each admits only the trivial automorphism.

The authors present an explicit construction of an isotope of the 2⁵-element semifield that contains a subsemifield of order 2².     

On the Hughes–Kleinfeld and Knuth’s semifields two-dimensional over a weak nucleus

In 1960 Hughes and Kleinfeld (Am J Math 82:389–392, 1960) constructed a finite semifield which is two-dimensional over a weak nucleus, given an automorphism σ of a finite field and elements with the property that has no roots in . In 1965 Knuth (J Algebra 2:182–217, 1965) constructed a further three finite semifields which are also two-dimensional over a weak nucleus, given the same parameter set . Moreover, in the same article, Knuth describes operations that allow one to obtain up to six semifields from a given semifield. We show how these operations in fact relate these four finite semifields, for a fixed parameter set, and yield at most five non-isotopic semifields out of a possible 24. These five semifields form two sets of semifields, one of which consists of at most two non-isotopic semifields related by Knuth operations and the other of which consists of at most three non-isotopic semifields.      

On a generalization of cyclic semifields

A new construction is given of cyclic semifields of orders q ²ⁿ , n odd, with kernel (left nucleus) and right and middle nuclei isomorphic to , and the isotopism classes are determined. Furthermore, this construction is generalized to produce potentially new semifields of the same general type that are not isotopic to cyclic semifields. In particular, a new semifield plane of order 4⁵ and new semifield planes of order 16⁵ are constructed by this method.     

A new semifield of order 2

In [G. Lunardon, Semifields and linear sets of PG(1,q^t), Quad. Mat., Dept. Math., Seconda Univ. Napoli, Caserta (in press)], G. Lunardon has exhibited a construction method yielding a theoretical family of semifields of order q²ⁿ,n>1 and n odd, with left nucleus F_qⁿ, middle and right nuclei both F_q² and center F_q. When n=3 this method gives an alternative construction of a family of semifields described in [N.L. Johnson, G. Marino, O. Polverino, R. Trombetti, On a generalization of cyclic semifields, J. Algebraic Combin. 26 (2009), 1-34], which generalizes the family of cyclic semifields obtained by Jha and Johnson in [V. Jha, N.L. Johnson, Translation planes of large dimension admitting non-solvable groups, J. Geom. 45 (1992), 87-104]. For n>3, no example of a semifield belonging to this family is known.In this paper we first prove that, when n>3, any semifield belonging to the family introduced in the second work cited above is not isotopic to any semifield of the family constructed in the former. Then we construct, with the aid of a computer, a semifield of order 2¹⁰ belonging to the family introduced by Lunardon, which turns out to be non-isotopic to any other known semifield.     

On Strong Isotopy of Dickson Semifields and Geometric Implications

Similarity of certain abelian collineation groups of a translation plane corresponds to strong isotopy of multiplication variations of a commutative semifield. Strong isotopy of Dickson semifields and their multiplication variations is characterized. The splitting of the isotopy class of a Dickson semifield over different types of basic fields (e. g. absolutely algebraic fields, number fields) into classes of strong isotopy is investigated. **     

Derived semifield planes of odd order admitting a dihedral group of affine homologies

Derived semifield planes of odd order admitting non-trivial involutorial affine homologies with more than one axis, are examined in detail, under the assumption that the group generated in the translation complement is dihedral. The whole structure of the semifields S coordinatizing such planes is determined. The class of the semifields S of dimension 4 over their centres is characterized.Dedicated to A. Barlotti on the occasion of his 65. birthday.Research partially supported by G.N.S.A.G.A. (C.N.R.)     

Extension of congruences on semirings of continuous functions

The lattices of congruences of semirings and semifields of continuous nonnegative functions over an arbitrary topological space are studied. It is proved that congruences of the semifield of continuous positive functions can be extended to congruences of the semiring of continuous nonnegative functions.     

Moduli spaces of local systems and higher Teichmüller theory

Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant under the action of the mapping class group of S. We prove that their transition functions are subtraction free. Thus we have positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their positive real points provide the two higher Teichmüller spaces related to G and S, while the points with values in the tropical semifields provide the lamination spaces. We define the motivic avatar of the Weil–Petersson form for one of these spaces. It is related to the motivic dilogarithm.     

Decompositions of Gelfand–Shilov kernels into kernels of similar class

We prove that any linear operator with a kernel in a Gelfand–Shilov space is a composition of two operators with kernels in the same Gelfand–Shilov space. We also give links on numerical approximations for such compositions. We apply these composition rules to establish Schatten–von Neumann properties for such operators.     

The BEL-rank of finite semifields

In this article we introduce the notion of the BEL-rank of a finite semifield, prove that it is an invariant for the isotopism classes, and give geometric and algebraic interpretations of this new invariant. Moreover, we describe an efficient method for calculating the BEL-rank, and present computational results for all known small semifields.     

Further results on planar DO functions and commutative semifields

It is proven that any Dembowski–Ostrom polynomial is planar if and only if its evaluation map is 2-to-1, which can be used to explain some known planar Dembowski–Ostrom polynomials. A direct connection between a planar Dembowski–Ostrom polynomial and a permutation polynomial is established if the corresponding semifield is of odd dimension over its nucleus. In addition, all commutative semifields of order 3⁵ are classified.