Description of polygonal regions by polynomials of bounded degree

We show that every (possibly unbounded) convex polygon P in \({\mathbb{R}^2}\) with m edges can be represented by inequalities p ₁ ≥ 0, . . ., p _n ≥ 0, where the p _i ’s are products of at most k affine functions each vanishing on an edge of P and n = n(m, k) satisfies \({s(m, k) \leq n(m, k) \leq (1+\varepsilon_m) s(m, k)}\) with s(m,k) ? max {m/k, log₂ m} and \({\varepsilon_m \rightarrow 0}\) as \({m \rightarrow \infty}\). This choice of n is asymptotically best possible. An analogous result on representing the interior of P in the form p ₁ > 0, . . ., p _n >  0 is also given. For k ≤ m/log₂ m these statements remain valid for representations with arbitrary polynomials of degree not exceeding k.     

Approximation by polynomials with bounded coefficients

Let θ be a real number satisfying 1<θ<2, and let A(θ) be the set of polynomials with coefficients in {0,1}, evaluated at θ. Using a result of Bugeaud, we prove by elementary methods that θ is a Pisot number when the set (A(θ)−A(θ)−A(θ)) is discrete; the problem whether Pisot numbers are the only numbers θ such that 0 is not a limit point of (A(θ)−A(θ)) is still unsolved. We also determine the three greatest limit points of the quantities , where C(θ) is the set of polynomials with coefficients in {−1,1}, evaluated at θ, and we find in particular infinitely many Perron numbers θ such that the sets C(θ) are discrete.     

Approximation of monomials by lower degree polynomials

Our topic is the uniform approximation ofx ^k by polynomials of degreen (n on the interval [–1, 1]. Our major result indicates that good approximation is possible whenk is much smaller thann ² and not possible otherwise. Indeed, we show that the approximation error is of the exact order of magnitude of a quantity,p _k,n , which can be identified with a certain probability. The numberp _k,n is in fact the probability that when a (fair) coin is tossedk times the magnitude of the difference between the number of heads and the number of tails exceedsn.     

ALTERNATION THEORY IN APPROXIMATION BY POLYNOMIALS HAVING BOUNDED COEFFICIENTS

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On conformal mapping of polygonal regions

We develop Kufarev's method for determining unknown parameters in the Schwarz-Christoffel integral in the case of conformal mapping of polygonal regions with boundary normalization.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 11, pp. 1484–1494, November, 1993.     

Permutation polynomials and orthomorphism polynomials of degree six

A classic paper of Dickson gives a complete list of permutation polynomials of degree less than 6 over arbitrary finite fields, and degree 6 over finite fields of odd characteristic. However, some published statements have hinted that Dicksonʼs classification might be incomplete in the degree 6 case. We uncover the reason for this confusion, and confirm the list of degree 6 permutation polynomials over all finite fields. Using this classification, we determine the complete list of degree 6 orthomorphism polynomials. Additionally, we note that a family of permutation polynomials from Dicksonʼs list provides counterexamples to a published conjecture of Mullen.     

On bounded polynomials in several variables

Mathematische Zeitschrift -     

Distribution of values of bounded generalized polynomials

A generalized polynomial is a real-valued function which is obtained from conventional polynomials by the use of the operations of addition, multiplication, and taking the integer part; a generalized polynomial mapping is a vector-valued mapping whose coordinates are generalized polynomials. We show that any bounded generalized polynomial mapping u: Z ^d  → R ^l has a representation u(n) = f(ϕ(n)x), n ∈ Z ^d , where f is a piecewise polynomial function on a compact nilmanifold X, x ∈ X, and ϕ is an ergodic Z ^d -action by translations on X. This fact is used to show that the sequence u(n), n ∈ Z ^d , is well distributed on a piecewise polynomial surface (with respect to the Borel measure on that is the image of the Lebesgue measure under the piecewise polynomial function defining ). As corollaries we also obtain a von Neumann-type ergodic theorem along generalized polynomials and a result on Diophantine approximations extending the work of van der Corput and of Furstenberg–Weiss.     

Construction of cubature formulas of degree eleven for symmetric planar regions,using orthogonal polynomials

Summary A method of constructing 28-point, 26-point and 25-point cubature formulas with polynomial precision 11 is given for planar regions and weight functions, which are symmetric in each variable. The nodes are computed as common zeros of a set of linearly independent orthogonal polynomials.Aspirant of the N.F.W.O., Belgium.     

Embedding nearly-spanning bounded degree trees

We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of maximum degree at most d on (1 − ε)n vertices, in terms of the expansion properties of G. As a result we show that for fixed d ≥ 2 and 0 < ε < 1, there exists a constant c = c(d, ε) such that a random graph G(n, c/n) contains almost surely a copy of every tree T on (1 − ε)n vertices with maximum degree at most d. We also prove that if an (n, D, λ)-graph G (i.e., a D-regular graph on n vertices all of whose eigenvalues, except the first one, are at most λ in their absolute values) has large enough spectral gap D/λ as a function of d and ε, then G has a copy of every tree T as above. Research supported in part by a USA-Israeli BSF grant, by NSF grant CCR-0324906, by a Wolfensohn fund and by the State of New Jersey. Research supported in part by USA-Israel BSF Grant 2002-133, and by grants 64/01 and 526/05 from the Israel Science Foundation. Research supported in part by NSF CAREER award DMS-0546523, NSF grant DMS-0355497, USA-Israeli BSF grant, and by an Alfred P. Sloan fellowship.     

Groups with bounded representation degree

Let Gbe a group and let K be a field of characteristic p> 0. If all irreducible representations of the group algebra K[G] have finite degree < n, then we show that G has a subgroup A with |G:A| bounded by a function of nand such that all the irreducible representations of K[A] have degree 1.     

Approximation of p-adic numbers by algebraic numbers of bounded degree

The approximation of p-adic numbers by algebraic numbers of bounded degree is studied. Results similar to those obtained by Wirsing and by Davenport and Schmidt in the real case are proved in the p-adic case. Unlike the real case the expected best exponent is not obtained when approximating by quadratic irrationals.