Uppers to zero in R[x] and almost principal ideals

Let R be an integral domain with quotient field K and f(x) a polynomial of positive degree in K[x]. In this paper we develop a method for studying almost principal uppers to zero ideals. More precisely, we prove that uppers to zero divisorial ideals of the form I = f(x)K[x] ∩ R[x] are almost principal in the following two cases:

• J, the ideal generated by the leading coefficients of I, satisfies J ^?1 = R.
• I ^?1 as the R[x]-submodule of K(x) is of finite type.

Furthermore we prove that for I = f(x)K[x] ∩ R[x] we have:

• I ^?1 ∩ K[x] = (I: K(x) I).
• If there exists p/q ∈ I ^?1 ? K[x], then (q, f) ≠ 1 in K[x]. If in addition q is irreducible and I is almost principal, then I′ = q(x)K[x] ∩ R[x] is an almost principal upper to zero.

Finally we show that a Schreier domain R is a greatest common divisor domain if and only if every upper to zero in R[x] contains a primitive polynomial.     

K 0 of hypersurfaces defined by x 1 2 + ... + x n 2 = ±1

Let k be a field of characteristic ≠ 2 and let Q _n,m (x ₁, ..., x _n , y ₁, ..., y _m ) = x ₁ ² +...+x _n ² ? (y ₁ ² +...+y _m ² ) be a quadratic form over k. Let R(Q _n,m ) = R _n,m = k[x ₁, ..., x _n , y ₁, ..., y _m ]/(Q _n,m ? 1). In this note we will calculate $\tilde K_0 \left( {R_{n,m} } \right)$ for every n,m ≥ 0. We will also calculate CH ₀(R _n,m ) and the Euler class group of R _n,m when k = ?.     

Solution of the Szili type conjugate problem on the roots of the laguerre polynomials

Let ?1<α≤0 and let $$L_n^{(\alpha )} (x) = \frac{1}{{n!}}x^{ - \alpha } e^x \frac{{d^n }}{{dx^n }}(x^{\alpha + n} e^{ - x} )$$ be the generalizednth Laguerre polynomial,n=1,2,… Letx ₁,x ₂,…,x _n andx*₁,x*₂,…,x* _n?1 denote the roots ofL _n ^(α) (x) andL _n ^(α)′ (x) respectively and putx*₀=0. In this paper we prove the following theorem: Ify ₀,y ₁,…,y _n ?1 andy ₁ ^′ ,…,y _n ^′ are two systems of arbitrary real numbers, then there exists a unique polynomialP(x) of degree 2n?1 satisfying the conditions $$\begin{gathered} P\left( {x_k^* } \right) = y_k (k = 0,...,n - 1) \hfill \\ P'\left( {x_k } \right) = y_k^\prime (k = 1,...,n). \hfill \\ \end{gathered} $$ .     

Compositions of Polynomials with Coefficients in a Given Field

Let F ⊂ K be fields of characteristic 0, and let K[x] denote the ring of polynomials with coefficients in K. Let p(x) = ∑_k = 0ⁿa_kx^k ∈ K[x], a_n ≠ 0. For p ∈ K[x]\F[x], define D_F(p), the F deficit of p, to equal n − max{0 ≤ k ≤ n : a_k∉F}. For p ∈ F[x], define D_F(p) = n. Let p(x) = ∑_k = 0ⁿa_kx^k and let q(x) = ∑_j = 0^mb_jx^j, with a_n ≠ 0, b_m ≠ 0, a_n, b_m ∈ F, b_j∉F for some j ≥ 1. Suppose that p ∈ K[x], q ∈ K[x]\F[x], p, not constant. Our main result is that p ° q ∉ F[x] and D_F(p ° q) = D_F(q). With only the assumption that a_nb_m ∈ F, we prove the inequality D_F(p ° q) ≥ D_F(q). This inequality also holds if F and K are only rings. Similar results are proven for fields of finite characteristic with the additional assumption that the characteristic of the field does not divide the degree of p. Finally we extend our results to polynomials in two variables and compositions of the form p(q(x, y)), where p is a polynomial in one variable.     

A note on the kernels of higher derivations

Let k ? k′ be a field extension. We give relations between the kernels of higher derivations on k[X] and k′[X], where k[X]:= k[x ₁,…, x _n ] denotes the polynomial ring in n variables over the field k. More precisely, let D = {D _n } _n=0 ^∞ a higher k-derivation on k[X] and D′ = {D′ _n } _n=0 ^∞ a higher k′-derivation on k′[X] such that D′ _m (x _i ) = D _m (x _i ) for all m ? 0 and i = 1, 2,…, n. Then (1) k[X] ^D = k if and only if k′[X] ^D′ = k′; (2) k[X] ^D is a finitely generated k-algebra if and only if k′[X] ^D′ is a finitely generated k′-algebra. Furthermore, we also show that the kernel k[X] ^D of a higher derivation D of k[X] can be generated by a set of closed polynomials.     

Characterization of a class of bivariate distribution functions

Let F_X,Y(x,y) be a bivariate distribution function and P_n(x), Q_m(y), n, m = 0, 1, 2,…, the orthonormal polynomials of the two marginal distributions F_X(x) and F_Y(y), respectively. Some necessary conditions are derived for the co-efficients c_n, n = 0, 1, 2,…, if the conditional expectation E[P_n(X) ∥ Y] = c_nQ_n(Y) holds for n = 0, 1, 2,…. Several examples are given to show the application of these necessary conditions.     

Analytic methods applied to a sequence of binomial coefficients

In previous work we have shown that the binomial coefficients C_n··kr, r are strongly logarithmically concave for 0?r?[n/(k+1)] and hence have at most a double maximum. Let r_{n, k} be the least integer at which this maximum occurs. Properties of {r_{n, k}}_{n, k} are best derived by introducing the polynomial family Q_k(x,y) defined by Q_k(x,y) = ∏^k+1_j=1 [1?(k+1)x+jy]?x∏^k_j=1[1?kx+jy]. It is shown that for each k there is a unique function η_k(y) defined on [0, 1/(k+1)] which is analytic in a neighbourhood of zero and which satisfies Q_k(η_k(y), y)=0. Setting η_k(0)=δ_k, η¹_k(0)=α_k we prove that r_rn,k = [nδ_k] o[nδ_k]+1 and further, that the number of times that r_m,k = [mδ_k]+1 for k+1? m ? n is asymptotically nα_k. Several other properties of α_k are derived, including 0<α_k <1/2.     

On the convex hull of the multiform numerical range

Let A ₁,…,A_m be nxn hermitian matrices. Definine

W(A ₁,…,A_m )={(xA₁x ^?,…xA_mx ^?):x?C ⁿ ,xx ^?=1}. We will show that every point in the convex hull of W(A ₁,…,A_m ) can be represented as a convex combination of not more than k(m,n) points in W(A ₁,…,A_m ) where k(m,n)=min{n,[√m]+δ _n ² _m+1}.     

On diophantine equations of the form {({\boldsymbol x}-{\boldsymbol a}_{\bf 1})({\boldsymbol x}-{\boldsymbol a}_{\bf 2}) \ldots ({\boldsymbol x}-{\boldsymbol a}_{\boldsymbol k}) + {\boldsymbol r} = {\boldsymbol y}^{ \boldsymbol n}}

Erd?s and Selfridge [3] proved that a product of consecutive integers can never be a perfect power. That is, the equation x(x?+?1)(x?+?2)...(x?+?(m???1))?=?y ⁿ has no solutions in positive integers x,m,n where m, n?>?1 and y?∈?Q. We consider the equation $$ (x-a_1)(x-a_2) \ldots (x-a_k) + r = y^n $$ where 0?≤?a ₁?<?a ₂?<???<?a _k are integers and, with r?∈?Q, n?≥?3 and we prove a finiteness theorem for the number of solutions x in Z, y in Q. Following that, we show that, more interestingly, for every nonzero integer n?>?2 and for any nonzero integer r which is not a perfect n-th power for which the equation admits solutions, k is bounded by an effective bound.     

Right inverses for linear, constant coefficient partial differential operators on distributions over open half spaces

Results of Hörmander on evolution operators together with a characterization of the present authors [Ann. Inst. Fourier, Grenoble 40, 619–655 (1990)] are used to prove the following: Let P ∈ ?[z₁,...,z _n ] and denote by P _m its principal part. If P ? P_m is dominated by P _m then the following assertions for the partial differential operators P(D) and P _m(D) are equivalent for N ∈ S ^n?1:

1. P(D) and/or P_m D)admit a continuous linear right inverse on C ^∞(H ₊(N)).
2. P(D) admits a continuous linear right inverse on C ^∞ (? ⁿ ) and a fundamental solution E ∈ C ^∞(?ⁿ) satisfying Supp $E \subset \overline {H - (N)} $

where H ₊(N) := {x ∈ ? ⁿ :±(x,N) τ; 0}.     

On the exactness of certain inequalities in approximation theory

We prove the following: for every sequence {F_v}, F_v ? 0, F_v > 0 there exists a functionf such that

1. E_n(f)?F_n (n=0, 1, 2, ...) and
2. A_kn^?k? _v=1 ⁿ v^k?1 F_v?1?Ω_k (f, n^?1) (n=1, 2, ...).

     

Small cycles and 2-factor passing through any given vertices in graphs

The theory of vertex-disjoint cycles and 2-factor of graphs has important applications in computer science and network communication. For a graph G, let σ ₂(G):=min?{d(u)+d(v)|uv ? E(G),u≠v}. In the paper, the main results of this paper are as follows:

1. Let k≥2 be an integer and G be a graph of order n≥3k, if σ ₂(G)≥n+2k?2, then for any set of k distinct vertices v ₁,…,v _k , G has k vertex-disjoint cycles C ₁,C ₂,…,C _k of length at most four such that v _i ∈V(C _i ) for all 1≤i≤k.
2. Let k≥1 be an integer and G be a graph of order n≥3k, if σ ₂(G)≥n+2k?2, then for any set of k distinct vertices v ₁,…,v _k , G has k vertex-disjoint cycles C ₁,C ₂,…,C _k such that:
   1. v _i ∈V(C _i ) for all 1≤i≤k.
   2. V(C ₁)∪???∪V(C _k )=V(G), and
   3. |C _i |≤4, 1≤i≤k?1.

Moreover, the condition on σ ₂(G)≥n+2k?2 is sharp.     

Application of the Szili method of interpolation on the roots of the Legendre polynomials

Let $$P_n (x) = \frac{{( - 1)^n }}{{2^n n!}}\frac{{d^n }}{{dx^n }}\left[ {(1 - x^2 )^n } \right]$$ be thenth Legendre polynomial. Letx ₁,x ₂,…,x _n andx*₁,x*₂,…,x* _n?1 denote the roots ofP _n (x) andP′ _n (x), respectively. Putx ₀=x*₀=?1 andx* _n =1. In this paper we prove the following theorem: Ify ₀,y ₁,…,y _n andy′ ₀,y′ ₁, …,y′ _n are two systems of arbitrary real numbers, then there exists a unique polynomialQ _2n+1(x) of degree at most 2n+1 satisfying the conditions $$Q_{2n + 1} (x_k^* ) = y_k and Q_{2n + 1}^\prime (x_k ) = y_k^\prime (k = 0,...,n).$$ .     

Oscillatory and asymptotic behavior of solutions of differential equations with deviating arguments

The oscillatory and asymptotic behavior of solutions of a class of nth order nonlinear differential equations, with deviating arguments, of the form (E, δ) L_nx(t) + δq(t) f(x[g₁(t)],…, x[g_m(t)]) = 0, where δ = ± 1 and L₀x(t) = x(t), L_kx(t) = a_k(t)(L_{k ? 1}x(t))^., $\text{k = 1, 2,…, n (}{}^{\mathrm{.}}\text{=}\text{d}\text{dt}\text{)}$, is examined. A classification of solutions of (E, δ) with respect to their behavior as t → ∞ and their oscillatory character is obtained. The comparisons of (E, 1) and (E, ?1) with first and second order equations of the form y^.(t) + c₁(t) f(y[g₁(t)],…, y[g_m(t)]) = 0 and (a_{n ? 1}(t)z^.(t))^. ? c₂(t) f(z[g₁(t)],…, z[g_m(t)]) = 0, respectively, are presented. The obtained results unify, extend and improve some of the results by Graef, Grammatikopoulos and Spikes, Philos and Staikos.     

Supplement to the Pál type (0; 0, 1) lacunary interpolation

Letx ₁, …,x _n be givenn distinct positive nodal points which generate the polynomial $$\omega _n (x) = \prod\limits_{i = 1}^n {(x - x_i )} .$$ Letx*₁, …,x* _n?1 be the roots of the derivativeω′ _n (x) and putx ₀=0. In this paper, the following theorem is proved: Ify ₀, …,y _n andy′₁, …,y′ _n?1 are arbitrary real numbers, then there exists a unique polynomialP _2n?1(x) of degree 2n?1 having the following interpolation properties: $$P_{2n - 1} (x_j ) = y_j (j = 0,...,n),$$ , $$P_{2n - 1}^\prime (x_j^* ) = y_j^\prime (j = 1,...,n - 1).$$ . This result gives the theoretical completion of the original Pál type interpolation process, since it ensures uniqueness without assuming any additional condition.     

Polynomial Cunningham chains

A sequence of prime numbers p₁,p₂,p₃,…, such that pi=2pi−1+? for all i, is called a Cunningham chain of the first or second kind, depending on whether ?=1 or −1 respectively. If k is the smallest positive integer such that 2pk+? is composite, then we say the chain has length k. It is conjectured that there are infinitely many Cunningham chains of length k for every positive integer k. A sequence of polynomials f₁(x),f₂(x),… in Z[x], such that f₁(x) has positive leading coefficient, each fi(x) is irreducible in Q[x] and fi(x)=xfi−1(x)+? for all i, is defined to be a polynomial Cunningham chain of the first or second kind, depending on whether ?=1 or −1 respectively. If k is the least positive integer such that fk+1(x) is reducible in Q[x], then we say the chain has length k. In this article, for polynomial Cunningham chains of both kinds, we prove that there are infinitely many chains of length k and, unlike the situation in the integers, that there are infinitely many chains of infinite length, by explicitly giving infinitely many polynomials f₁(x), such that fk+1(x) is the only term in the sequence that is reducible.     

Tight bounds for rational sums of squares over totally real fields

Let K be a totally real Galois number field. Hillar proved that if f ∈ ?[x ₁, ..., x _n ] is a sum of m squares in K[x ₁, ..., x _n ], then f is a sum of N(m) squares in ?[x ₁, ..., x _n ], where N(m) ≤ 2^[K:?]+1 · $ \left( {_2^{[K:\mathbb{Q}] + 1} } \right) $ · 4m. We show in fact that N(m) ≤ m + 4 $ \left\lceil {\tfrac{m} {4}} \right\rceil $ ([K: ?] ? 1), our proof being constructive too. Moreover, we give some examples where this bound is sharp, for instance in the case of quadratic extensions. We also extend our results to the setting of non-commutative polynomials over ?.     

A combinatorial problem involving graphs and matrices

In this paper we discuss a combinatorial problem involving graphs and matrices. Our problem is a matrix analogue of the classical problem of finding a system of distinct representatives (transversal) of a family of sets and relates closely to an extremal problem involving 1-factors and a long standing conjecture in the dimension theory of partially ordered sets. For an integer n ?1, let n denote the n element set {1,2,3,…, n}. Then let A be a k×t matrix. We say that A satisfies property P(n, k) when the following condition is satisfied: For every k-taple (x₁,x₂,…,x_k?n^k there exist k distinct integers j₁,j₂,…,j_k so that x_i= a_ii for i= 1,2,…,k. The minimum value of t for which there exists a k × t matrix A satisfying property P(n,k) is denoted by f(n,k). For each k?1 and n sufficiently large, we give an explicit formula for f(n, k): for each n?1 and k sufficiently large, we use probabilistic methods to provide inequalities for f(n,k).     

Некоторые точные инт егральные оценки про изводных рациональных и алгеб раических функций. Пр иложения

Exact estimates are obtained for integrals of absolute values of derivatives and gradients, for integral moduli of continuity and for major variations of piecewise algebraic functions (in particular, for polynomials, rational functions, splines, etc.). These results are applied to the problems of approximation theory and to the estimates of Laurent and Fourier coefficients. Typical results:

1. IfE is a measurable subset of the circle or of a line in thez-plane andR(z) is a rational function of degree ≦n, ¦R(z)¦≦ (z∈E), then ∝_E ¦R′(z)¦dz¦≦ 2πn; the latter estimate is exact forn=0, 1, ... and everyE with positive measure;
2. Iff(x ₁,x ₂, ...,x _m) is a real valued piecewise algebraic function of order (n, k) on the unit ballD?R ^m (in particular, a real valued rational function of order ≦n), and ¦f¦≦1 onD, then ∝_D¦gradf¦dx≦2π ^m/2n/Π(m/2); herem≧1, n≧0, 1≦k<∞;
3. LetE=Π={z∶¦z¦=1}, and letc _m(R) be the mth Laurent coefficient ofR onΠ,C _m(n)=sup{¦cm(R)¦}, where sup is taken over allR from 1), then 1/7 min {n/¦m¦, 1} ≦C _m(n) ≦ min {n/¦m¦, 1}.

     

New families of graceful banana trees

Consider a family of stars. Take a new vertex. Join one end-vertex of each star to this new vertex. The tree so obtained is known as abanana tree. It is proved that the banana trees corresponding to the family of stars

1. (K_1,1, K_1,2,…, K_{1,t ?1}, (α + l) K_1,t, K_{1,t + 1}, …, K_1,n), α ? 0
2. (2K_1,1, 2K_1,2,…, 2K_{1,t? 1}, (α + 2)K_1,t, 2K_{1,t + 1, …}, 2K_1,n), 0 ? α <t and
3. (3K_1,t, 3K_1,2, …, 3K_1,n) are graceful.