Polytope bounds on multivariate value sets

We improve the upper bounds for the cardinality of the value set of a multivariable polynomial map over a finite field using the polytope of the polynomial. This generalizes earlier bounds only dependent on the degree of a polynomial.     

On mean approximation by polynomials with gaps on Caratheodory sets

The paper presents some new results on the possibility of approximation by polynomials with gaps. The approximations are done in the norm of the space L _p , 1 ≤ p < + ∞, on the Caratheodory sets in the complex plane. The lacunary versions of some results by Farell—Markushevich, S. Sinanian, A. L. Shahinian are obtained (Theorems 1, 3, 5). Similar approximations by the real parts of lacunary polynomials are given (Theorems 2, 4, 6). Dedicated to the memory of academician S. N. Mergelyan     

Recursively enumerable sets of polynomials over a finite field

We prove that a relation over is recursively enumerable if and only if it is Diophantine over . We do this by first constructing a model of in , where n is represented by Zⁿ. In a second step, we show that it suffices to eliminate a bounded universal quantifier. Then finally, the hardest part of the proof is to show that we can eliminate this quantifier.     

A refinement of multivariate value set bounds

Over finite fields, if the image of a polynomial map is not the entire field, then its cardinality can be bounded above by a significantly smaller value. Earlier results bound the cardinality of the value set using the degree of the polynomial, but more recent results make use of the powers of all monomials.In this paper, we explore the geometric properties of the Newton polytope and show how they allow for tighter upper bounds on the cardinality of the multivariate value set. We then explore a method which allows for even stronger upper bounds, regardless of whether one uses the multivariate degree or the Newton polytope to bound the value set. Effectively, this provides improvement of a degree matrix-based result given by Zan and Cao, making our new bound the strongest upper bound thus far.     

Recursively enumerable sets of polynomials over a finite field are Diophantine

We construct a Diophantine interpretation of over . Using this together with a previous result that every recursively enumerable (r.e.) relation over is Diophantine over , we will prove that every r.e. relation over is Diophantine over . We will also look at recursive infinite base fields , algebraic over . It turns out that the Diophantine relations over are exactly the relations which are r.e. for every recursive presentation.     

Graphs with few total dominating sets

We give a lower bound for the number of total dominating sets of a graph together with a characterization of the extremal graphs, for trees as well as arbitrary connected graphs of given order. Moreover, we obtain a sharp lower bound involving both the order and the total domination number, and characterize the extremal graphs as well.     

Polynomials and limited sets

We prove that scalar-valued polynomials are weakly continuous on limited sets and that, as in the case of linear mappings, every -valued polynomial maps limited sets into relatively compact ones. We also show that a scalar-valued polynomial whose derivative is limited is weakly sequentially continuous.

     

Spherical two-distance sets

A set S of unit vectors in n-dimensional Euclidean space is called spherical two-distance set, if there are two numbers a and b so that the inner products of distinct vectors of S are either a or b. It is known that the largest cardinality g(n) of spherical two-distance sets does not exceed n(n+3)/2. This upper bound is known to be tight for n=2,6,22. The set of mid-points of the edges of a regular simplex gives the lower bound L(n)=n(n+1)/2 for g(n).In this paper using the so-called polynomial method it is proved that for nonnegative a+b the largest cardinality of S is not greater than L(n). For the case a+b<0 we propose upper bounds on |S| which are based on Delsarte's method. Using this we show that g(n)=L(n) for 6<n<22, 23<n<40, and g(23)=276 or 277.     

Irreducible polynomials over a finite field

Dedicated to the memory of A. I. Mal'tsev.     

The topology of Julia sets for polynomials

We prove that wandering components of the Julia set of a polynomial are singletons provided each critical point in a wandering Julia component is non-recurrent. This means a conjecture of Branner-Hubbard is true for this kind of polynomials     

On scrambled sets and a theorem of Kuratowski on independent sets

The measure of scrambled sets of interval self-maps was studied by many authors, including Smítal, Misiurewicz, Bruckner and Hu, and Xiong and Yang. In this note, first we introduce the notion of ``-chaos" which is related to chaos in the sense of Li-Yorke, and we prove a general theorem which is an improvement of a theorem of Kuratowski on independent sets. Second, we apply the result to scrambled sets of higher dimensional cases. In particular, we show that if a map of the unit -cube is -chaotic on , then for any there is a map such that and are topologically conjugate, and has a scrambled set which has Lebesgue measure 1, and hence if , then there is a homeomorphism with a scrambled set satisfying that is an -set in and has Lebesgue measure 1.

     

Integer valued polynomials over a number field

A number field is called a Pólya field if the module of integer valued polynomials over that field is generated by (f_i) _i=0 over the ring of integers, with deg(f_i)=i, i=0, 1, 2,... In this paper bounds on the class numbers and on the number of ramified primes in Pólya fields are derived.     

On a problem of Byrnes concerning polynomials with restricted coefficients, II

As in the earlier paper with this title, we consider a question of Byrnes concerning the minimal length of a polynomial with all coefficients in which has a zero of a given order at . In that paper we showed that for all and showed that the extremal polynomials for were those conjectured by Byrnes, but for that rather than . A polynomial with was exhibited for , but it was not shown there that this extremal was unique. Here we show that the extremal is unique. In the previous paper, we showed that is one of the 7 values or . Here we prove that without determining all extremal polynomials. We also make some progress toward determining . As in the previous paper, we use a combination of number theoretic ideas and combinatorial computation. The main point is that if is a primitive th root of unity where is a prime, then the condition that all coefficients of be in , together with the requirement that be divisible by puts severe restrictions on the possible values for the cyclotomic integer .