Nonconstant Polynomial Research Articles

The algebraic hypercenter of an algebra over a commutative ring

In [5], Herstein shows that if x is an element of a ring R containing no nil ideals, and for each y E R there exists a positive integer n such that xy” = y’k, then x is in the center of A. Bergen and Herstein [ 1 ] extend this result to algebras over a field in the following way. Let R be an algebra over a field F such that R contains no ideals that consist only of elements algebraic over F. If x is an element of R, and for each y E R there exists a nonconstant polynomial pX, J t) E F[ t] such that xpX, J y) = pX, J y )x, then x is in the center of R. We consider algebras R over a commutative ring C. We do not assume that R contains a unit element. However, we assume that C has a unit element 1 and that lr = r for each r E R. For an ideal 1 of R, define A,(1) to be the set of elements x E R such that for each y E I there exists a monk polynomial of positive degree P,,~( t) E C[ t], such that XJP, Y( y) = pX,,( y )x. It is clear that A,(I) is a subring of R. The ring A,(R) is called the algebraic hypercenter of R (over C). We write AR(J) as A(I) when it is clear which algebra R is being considered. Throughout this paper, a polynomial is assumed to have degree greater than zero. An ideal I of R will be called a C-integral ideal or an ideal integral over C, if for each y E 1, there exists a monk polynomial p,(t) E C[t] such that p,(y) = 0. The ideal I will be called a C-algebraic ideal or dgebraic over C, if for each y E I there exists a polynomial p,(t) E C[t] that is not necessarily manic, such that p,(y) = 0. Suppose that B is a subalgebra of R. Then R is said to be C-integral over B, if for each x E R there exists a manic polynomial f,(t) E C[ t] such that f,(x) 6 B247 0021-8693/90 fE3.00

On permutability of periodic entire functions

On donne les fonctions permutables avec une fonction entiere periodique de type exponentiel

ON THE INTEGRABILITY OF HAMILTONIAN SYSTEMS WITH TORAL POSITION SPACE

This paper considers the problem on the complete integrability of a Hamiltonian system with a toral position space, with Euclidean kinetic energy and a small analytic potential. Necessary integrability conditions are found in the case when the potential is a trigonometric polynomial. These conditions are also necessary conditions of existence of additional first integrals, polynomial in the momenta (with no assumption on the smallness of the potential). The proofs are based on a detailed analysis of the classical scheme of perturbation theory. The general results are applied to the study of the complete integrability of the well-known problem on the motion of n points along a line with interaction potential. In particular, the nonintegrability of the open chain of interactions of particles is proved for n > 2; the periodic chain is nonintegrable with the additional condition that the potential be a nonconstant trigonometric polynomial. Conditions for complete integrability of the generalized nonperiodic Toda chain are discussed.Bibliography: 17 titles.

2 × 2 monic irreducible matrix polynomials

A complex polynomial is known to be irreducible (i.e. to have no proper non-constant polynomial divisors) only if it is of first degree. Over the reals, we get irreducibles of degrees 1 and 2 only. Obviously this is not the case for a matrix valued polynomial. In this note the 2 × 2 monic-irreducibles are found, and their Jordan structure is examined. Both The real and complex cases are considered apart from linear ones, the only irreducibles over have the form (λ α) m I + p(λ)A (see below). Over we also get irreducible diagonal quadratics and irreducibles of the form .

On the probability that the values of m polynomials have a given g.c.d.

Let m ≥ 2, f 1, f 2,…, f m arbitrary nonconstant polynomials with integer coefficients, and h a positive integer. Let M( x; f 1, f 2,…, f m ; h) denote the number of m-tuples 〈 x 1, x 2,…, x m 〉 of positive integers such that x i ≤ x for 1 ≤ i ≤ m and ( f 1( x 1), f 2( x 2),…, f m ( x m )) = h and let d( f 1, f 2,…, f m ; h) = lim x → ∞ x − m M( x; f 1, f 2,…, f m ; h). Recently, R. N. Buttsworth ( J. Number. Theory 12 (1980) , 487–498) proved that d(f 1, f 2, …, f m, h) = 1 h m ∑ n=1 x μ(n) Π i=1 m ϱ i(nh) n m , (∗) where ϱ i ( n) denotes the number of solutions (mod n) of the congruence f i ( x) ≡ 0 (mod n). Unfortunately, his proof contains mistakes. In this paper, by a different method, (∗) is established. In fact, an asymptotic formula with an error term for M( x; f 1, f 2, f m ; h) is obtained.

On the sums Σ n ⩽ xA( f( n)) and Σ p ⩽ xA( f( p))

For a complex number s and an arithmetical function α, we write A( n) = Σ dδ = n α( d) δ s and A ∗(n) = A(n) n −s . In this paper, we prove asymptotic formulae for the sums Σ n⩽ x A( f( n)), Σ p⩽ x A( f( p)), Σ n⩽x A ∗(f(n)) , and Σ p⩽x A ∗(f(p)) , where Re s > 0, f is a nonconstant polynomial with integer coefficients, and α satisfies a growth condition. Several illustrations are given which incidentally refine results due to R. Bellman, W. Schwarz, and W. A. Webb.

Factoring Finite Factor Rings

The most descriptive way to represent a finite Abelian group is as a direct product of cyclic groups. In this form, many important characteristics such as order, exponent, and rank are immediately evident. For instance, consider the group G of units of the ring of integers modulo 180. With this description of G, very little about its structure is readily apparent. On the other hand, if we are told that G is isomorphic to Z12 X Z2 X Z2 (see [9] or [11, p. 46]), then we may observe that G has order 48, exponent 12, and rank 3. More generally, one could ask for the structure of the group of units of any commutative ring R. When R is finite, we know from the Fundamental Theorem of Finite Abelian Groups the beautiful fact that the group of units of R, U(R), is isomorphic to a direct product of cyclic groups. So, it is natural to look for a method to express the group of units of any finite commutative ring as a direct product of cyclic groups. This problem has not been solved in general, but the solution to the special case in which R is a finite field is well known. In this case, U(R) is cyclic [8, p. 405]. Other familiar finite rings are those of the form R = F[x]/h(x)>, where F is a finite field and h(x) is a nonconstant polynomial in F[x]. For example, consider R = Z7[X]/((X + 4)8(X2 + 4)(X3 + 3)2). What is U(R) in this case? It turns out that a nice blend of elementary ring theory and group theory gives an algorithm which yields the answer:

Polynomials on Affine Manifolds

For a closed affine manifold $M$ of dimension $m$ the developing map defines an open subset $D(\tilde M) \subset {{\mathbf {R}}^m}$. We show that $D(\tilde M)$ cannot lie between parallel hyperplanes. When $m \le 3$ we show that any nonconstant polynomial $p:{{\mathbf {R}}^m} \to {\mathbf {R}}$ is unbounded on $D(\tilde M)$. If $D(\tilde M)$ lies in a half-space we show $M$ has zero Euler characteristic. Under various special conditions on $M$ we show that $M$ has no nonconstant functions given by polynomials in affine coordinates.

On Lucas-Sets for Vector-Valued Abstract Polynomials in K-inner Product Spaces

The fact that Rolle's theorem on critical points of a real differentiable function does not hold in general for analytic functions of a complex variable raises a natural question [7, p. 21] as to whether or not it can be generalized to polynomials, the simplest subclass of analytic functions. While attempting to answer this and other related problems in [4, 5, 6], Lucas proved that all the critical points of a nonconstant polynomial f lie in the convex hull of the set of zeros of f (see Theorem (6.1) in [7]). Walsh [12] has shown that Lucas’ theorem is equivalent to the following result [7, Theorem (6.2)], namely: Any convex circular region which contains all the zeros of a polynomial f also contains all the zeros of its derivative f′.

Polynomial forms on affine manifolds

An affine manifold is a differentiable manifold without boundary together with a maximal atlas of coordinate charts such that all coordinate changes extend to affine automorphisms of R These distinguished charts are called affine coordinate systems. Throughout this paper M denotes a connected affine manifold of dimension n^l. We write E for R. A tensor (field) on M is called polynomial if in all affine coordinate systems its coefficients are polynomial functions in n variables. In particular a real-valued function on M may be polynomial. It is unknown whether there exists any compact affine manifold admitting a nonconstant polynomial function. The main purpose of this paper is to prove that for certain classes of affine manifolds there is no such function. These results are then applied to demonstrate that certain polynomial forms must also vanish. For related results, see Fried, Goldman, and Hirsch [2], Fried [1], [6], and [5].

Proof of the Fundamental Theorem of Algebra

(1981). Proof of the Fundamental Theorem of Algebra. The American Mathematical Monthly: Vol. 88, No. 4, pp. 253-256.

Van der corput’s difference theorem

We obtain a sufficient condition for a subsetH of positive integers to satisfy that the equidistribution (mod 1) of the sequences (u n+h − u n; n=1, 2, ···) for allh ∈H implies the equidistribution of (u n). Our condition is satisfied, for example, for the following sets: (1)H={n − m; n ∈ I, m ∈ I, n>m}, whereI is any infinite subset of integers; (2)H={| ψ (n)|; ψ(n)≠0,n ∈ Z}, where ψ is a nonconstant polynomial with integral coefficients having at least one integral zero (modq) for allq=2, 3, ···; (3)H={p+1;p is a prime} andH={p − 1;p is a prime}.

Algebraically closed noncommutative polynomial rings

Let R be a noncommutative polynomial ring over the division ring K where K has center F. Then R = K[x,σ,D]where σ is a monomorphism of K and D is a σ-derivaton K. R is called dimension finite if (K: Fσ)<∞ and (K: FD)<∞ where Fσ is the subfield of F fixed under σand FD is the subfied of F of D-constants. R is algebraically closed if every nonconstant polynomial in Rfactors completely into linear factors. The algebraically closed dimension finite polynomial rings are determined. s done by reducing the problem to two classes: skew polynomial rings and differential polynomial rings. Examples algebraically closed polynomial rings which are not dimensfinite are given.

Further results on prime entire functions

Let H denote the set of all the entire functions f ( z ) f(z) of the form: f ( z ) ≡ h ( z ) e p ( z ) + k ( z ) f(z) \equiv h(z){e^{p(z)}} + k(z) where p ( z ) p(z) is a nonconstant polynomial of degree m, and h ( ≢ 0 ) h(\nequiv \;0) , k ( ≢ k(\nequiv constant) are two entire functions of order less than m. In this paper, a necessary and sufficient condition for a function in H to be a prime is established. Several generalizations of known results follow. Some sufficient conditions for primeness of various subclasses of H are derived. The methods used in the proofs are based on Nevanlinna’s theory of meromorphic functions and some elementary facts about algebraic functions.

The Fix-Points and Factorization of Meromorphic Functions

In this paper, we use the Nevanlinna theory of meromorphic functions and a result of Goldstein to generalize some known results in factorization and fixpoints of entire functions. Specifically, we prove (1) If $f$ and $g$ are nonlinear entire functions such that $f(g)$ is transcendental and of finite order, then $f(g)$ has infinitely many fix-points. (2) If $f$ is a polynomial of degree $\geqq 3$, and $g$ is an arbitrary transcendental meromorphic function, then $f(g)$ must have infinitely many fix-points. (3) Let $p(z),q(z)$ be any nonconstant polynomials, at least one of which is not $c$-even, and let $a$ and $b$ be any constants with $a$ or $b \ne 0$. Then $h(z) = q(z)\exp (a{z^2} + bz) + p(z)$ is prime.

The fix-points and factorization of meromorphic functions

In this paper, we use the Nevanlinna theory of meromorphic functions and a result of Goldstein to generalize some known results in factorization and fixpoints of entire functions. Specifically, we prove (1) If f f and g g are nonlinear entire functions such that f ( g ) f(g) is transcendental and of finite order, then f ( g ) f(g) has infinitely many fix-points. (2) If f f is a polynomial of degree ≧ 3 \geqq 3 , and g g is an arbitrary transcendental meromorphic function, then f ( g ) f(g) must have infinitely many fix-points. (3) Let p ( z ) , q ( z ) p(z),q(z) be any nonconstant polynomials, at least one of which is not c c -even, and let a a and b b be any constants with a a or b ≠ 0 b \ne 0 . Then h ( z ) = q ( z ) exp ⁡ ( a z 2 + b z ) + p ( z ) h(z) = q(z)\exp (a{z^2} + bz) + p(z) is prime.

On the Fundamental Theorem of Algebra

(1967). On the Fundamental Theorem of Algebra. The American Mathematical Monthly: Vol. 74, No. 5, pp. 485-497.

On Herstein’s Theorem Concerning Three Fields

Let L &gt; K ≧ Φ, LK, be three fields such that: (1) L/K is not purely inseparable, and (2) L/Φ is transcendental. Then Herstein’s theorem [2] asserts the existence of u ∈ L such that f(u) ∉ K for every non-constant polynomial f(X) ∈ Φ[X].