On polynomial interpolation of two variables

Polynomial interpolation of two variables based on points that are located on multiple circles is studied. First, the poisedness of a Birkhoff interpolation on points that are located on several concentric circles is established. Second, using a factorization method, the poisedness of a Hermite interpolation based on points located on various circles, not necessarily concentric, is established. Even in the case of Lagrange interpolation, this gives many new sets of poised interpolation points.     

A discriminant criteria for reducibility of a polynomial

Letp(w) be a polynomial over a domainK. Ifp splits to linear factors then the discriminantD(p) is a square inK. In this paper we state an additional condition on the roots ofp which together with the discriminant condition imply the splitting ofp in case thatK=C [z] or Z. Some extensions are also discussed. Visiting Lady Davis Research Professor at the Hebrew University of Jerusalem, Fall Semester, 1984. Supported in part by US-Israel Binational Science Foundation grant 3225/84.     

Images of locally finite derivations of polynomial algebras in two variables

In this paper we show that the image of any locally finite k-derivation of the polynomial algebra k[x,y] in two variables over a field k of characteristic zero is a Mathieu subspace. We also show that the two-dimensional Jacobian conjecture is equivalent to the statement that the image of every k-derivation D of k[x,y] such that and is a Mathieu subspace of k[x,y].     

Extensions of thep-adic fields generated by the roots of a single polynomial in two variables

One of the generalizations of the Hilbert's Irreducibility Theorem states that if F(X, Y) is irreducible in Q _p [X, Y] then, for almost every n N, F(n, Y) is irreducible in Q _p [Y]. Let E{p, d} be the set of all the extensions of Q _p of degree d. We shall prove that for each subset S E{p, d}there exist polynomials F(X, Y) such that the set of extensions of Q _p generated by the roots of F (n, Y)is exactly S. Moreover we shall prove that all the functions f are induced by polynomials F(X) Z[X].M. S. C. N. 11S05, (11S15).The author would like to thank Prof. Roberto Dvornicich for several useful conversations and advices.     

Poisson (co)homology of truncated polynomial algebras in two variables

We study the Poisson (co)homology of the algebra of truncated polynomials in two variables viewed as the semi-classical limit of a quantum complete intersection studied by Bergh and Erdmann. We show in particular that the Poisson cohomology ring of such a Poisson algebra is isomorphic to the Hochschild cohomology ring of the corresponding quantum complete intersection. To cite this article: S. Launois, L. Richard, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Cycles of polynomial mappings in two variables over rings of integers in quadratic fields

We find all possible cycle-lengths for polynomial mappings in two variables over rings of integers in quadratic extensions of rationals.     

External rays for polynomial maps of two variables associated with Chebyshev maps

The dynamics of regular polynomial endomorphisms of two variables is investigated. Especially, the landing points of the external rays for maps associated with Chebyshev maps are completely characterized.     

The depth of the Jacobian ring of a homogeneous polynomial in three variables

The question as to whether the Jacobian ideal of an irreducible projective plane curve always admits an irrelevant component has been going around for some years. One shows that a curve will satisfy this if it has only ordinary nodes or cusps, while an example is given of a family of sextic curves whose respective Jacobian ideals are saturated. The connection between this problem and the theory of homogeneous free divisors in three variables is also pointed out, so the example gives a family of Koszul-free divisors.

     

On pairs of commuting derivations of the polynomial ring in one or two variables

It is well known that each pair of commuting linear operators on a finite dimensional vector space over an algebraically closed field has a common eigenvector. We prove an analogous statement for derivations of k[x] and k[x,y] over any field k of zero characteristic. In particular, if D₁ and D₂ are commuting derivations of k[x,y] and they are linearly independent over k, then either (i) they have a common polynomial eigenfunction; i.e., a nonconstant polynomial f∈k[x,y] such that D₁(f)=λf and D₂(f)=μf for some λ,μ∈k[x,y], or (ii) they are Jacobian derivations

     

Remarks on polynomial expansions of solutions of the heat equation in two space variables

Results on polynomial expansions of analytic solutions of the heat equation can be used for the discussion of the continuation of analytic solutions. A system of polynomial solutions introduced by Col ton and Wimp [3] is found good for such investigations in the two dimensional case. A Banach scales approach is the base for the results of the present paper     

The global dimension of polynomial categories in partially commuting variables

We study the global dimension of the category of objects of an abelian category carrying an action of a free partially commutative monoid. We calculate this dimension in the case that the abelian category has infinite coproducts and enough projectives. Previously the author solved the same problem for abelian categories with exact coproducts.     

Primes are nonnegative values of a polynomial in 10 variables

A polynomial with integer coefficients of 10 variables is constructed, whose set of all nonnegative values (for positive integer values of the variables) is precisely the set of all prime numbers.Translated by Louise Guy and James P. Jones, The University of Calgary, Calgary, CanadaTranslated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 68, pp. 62–82, 1977.     

On the best approximation of periodic functions of two variables by polynomial splines

We consider the problem of the best approximation of periodic functions of two variables by a subspace of splines of minimal defect with respect to a uniform partition.     

Cycles of polynomial mappings in several variables

Supported by the KBN-grant 2-1037-91-01     

Adaptive order polynomial algorithm in a multiwavelet representation scheme

We have developed a new strategy to reduce the storage requirements of a multivariate function in a multiwavelet framework. We propose that alongside the commonly used adaptivity in the grid refinement one can also vary the order of the representation k as a function of the scale n. In particular the order is decreased with increasing refinement scale. The consequences of this choice, in particular with respect to the nesting of scaling spaces, are discussed and the error of the approximation introduced is analyzed. The application of this method to some examples of mono- and multivariate functions shows that our algorithm is able to yield a storage reduction up to almost 60%. In general, values between 30 and 40% can be expected for multivariate functions. Monovariate functions are less affected but are also much less critical in view of the so called “curse of dimensionality”.