Real Polynomials Of Degree Research Articles

Counting zeros of generalised polynomials: Descartes’ rule of signs and Laguerre’s extensions

One of the bedrock theorems of mathematics is the statement that a real polynomial of degree n has at most n real zeros. Probably the best-known proof is the algebraic one, by factorisation. But there is also a pleasant analytic proof, by deduction from Rolle’s theorem.A slightly different question is how many positive zeros a polynomial has. Here the basic result is known as ‘Descartes′ rule of signs’. It says that the number of positive zeros is no more than the number of sign changes in the sequence of coefficients. Descartes included it in his treatise La Géométrie which appeared in 1637. It can be proved by a method based on factorisation, but, again, just as easily by deduction from Rolle’s theorem.

On a nonlinear dispersive equation with time-dependent coefficients

As a first step, we consider an evolution linear problem, the symbol of which is a real polynomial of degree three with time-dependent coefficients. We get for this problem smoothing effects known when these coefficients are constant. In particular, by using the theory of Calderón-Zygmund operators and the David and Journé T1 Theorem, we establish a local smoothing effect on the solution of the linear problem. In a second step, we study a nonlinear dispersive equation the linear part of which is the one studied above. We use the previous smoothing properties and a regularization method to establish that the Cauchy problem is locally well-posed in the Sobolev spaces $H^s(\mathbb R)$ for $s>3/4$.

Integral Geometry and Real Zeros of Thue–Morse Polynomials

We study the average number of intersecting points of a given curve with random hyperplanes in an n-dimensional Euclidean space. As noticed by A. Edelman and E. Kostlan, this problem is closely linked to finding the average number of real zeros of random polynomials. They showed that a real polynomial of degree n has on average log n + O(1) real zeros (M. Kac's theorem). This result leads us to the following problem: given a real sequence (α k ) k ∊N, study the average where ρ(fn) is the number of real zeros of fn(X) = α0+α1X+ … + αnXn. We give theoretical results for the Thue—Morse polynomials and numerical evidence for other polvnomials

On Certain Mean Values of Polynomials on the Unit Interval

For any continuous function f:[−1, 1]↦C and any p∈(0, ∞), let ‖f‖p≔(2−1∫1−1|f(x)|pdx)1/p; in addition, let ‖f‖∞≔max−1⩽x⩽1|f(x)|. It is known that if f is a polynomial of degree n, then for all p>0,‖f‖∞⩽Cpn2/p‖f‖p,where Cp is a constant depending on p but not on n. In this result of Nikolskiı (1951), which was independently obtained by Szegö and Zygmund (1954), the order of magnitude of the bound is the best possible. We obtain a sharp version of this inequality for polynomials not vanishing in the open unit disk. As an application we prove the following result. If f is a real polynomial of degree n such that f(−1)=f(1)=0 and f(z)≠0 in the open unit disk, then for p>0 the quantity ‖f′‖∞/‖f‖p is maximized by polynomials of the form c(1+x)n−1(1−x), c(1+x)(1−x)n−1, where c∈R\\{0}. This extends an inequality of Erdős (1940).

Uniform decay estimates for a class of oscillatory integrals and applications

One dimensional oscillatory integrals of the type $\int^\infty _0 \xi ^\alpha \rm\exp \big[it(p(\xi )-\xi x)\big]\rm {d}\xi $ are considered, where $p(\xi )$ is a real polynomial of degree $m\geq 3$. Long-time decay and global smoothing estimates are established, as well as short-time behavior as $t\to 0$. The results are applied to the fundamental solutions of a class of linearized Kadomtsev-Petviashvili equations with higher dispersion

The index of [formula omitted

Let f(x, y) be a real polynomial of degree d with isolated critical points, and let i be the index of grad f around a large circle containing the critical points. An elementary argument shows that | i|⩽ d−1. In this paper we show that i⩽ max{1, d−3} . We also show that if all the level sets of f are compact, then i=1, and otherwise |i|⩽d R −1 where d R is the sum of the multiplicities of the real linear factors in the homogeneous term of highest degree in f. The technique of proof involves computing i from information at infinity. The index i is broken up into a sum of components i p, c corresponding to points p in the real line at infinity and limiting values c∈ R ∪{∞} of the polynomial. The numbers i p, c are computed in three ways: geometrically, from a resolution of f(x, y) , and from a Morsification of f(x, y) . The i p, c also provide a lower bound for the number of vanishing cycles of f(x, y) at the point p and value c.

Is the Polar Decomposition Finitely Computable?

The polar decomposition of a square matrix is a major step toward the singular value decomposition, and is an important device in its own right. Singular values of a matrix A, being the eigenvalues for a matrix closely related to A, generally cannot be computed by a finite process if only arithmetic operations and radicals are allowed. However, this consideration alone does not prove that the polar decomposition is not finitely computable. The problem of finite computability of the polar decomposition is not settled in this paper, but we do show it to be equivalent to the following simpler-looking problem. Suppose f is a real polynomial of degree $n > 4$, and all the roots of f are distinct positive numbers. Denote by g a polynomial of the same degree whose zeros are the positive square roots of the zeros of f. Can this polynomial g always be computed finitely for a given polynomial f? In the Appendix we discuss one nontrivial situation where the polar decomposition can indeed be computed finitely.

Where Not to Find the Critical Points of a Polynomial--Variation on a Putnam Theme

(1995). Where Not to Find the Critical Points of a Polynomial—Variation on a Putnam Theme. The American Mathematical Monthly: Vol. 102, No. 2, pp. 155-158.

Iteration and the Zeros of the Second Derivative of a Meromorphic Function

Suppose that f = (f1/f2)ep where the fi are real entire functions of order less than n with only finitely many non-real zeros and P is a real polynomial of degree n. Suppose that f1, or f2 is a polynomial. It is shown that fn has at least n − 2 distinct non-real zeros. The proof is based on the iteration of transcendental meromorphic functions.

Cardinal interpolation by polynomial splines: Interpolation of data with exponential growth

Let m ∈ N and define S m to be the class of functions ƒ ∈ C m− 1 ( R which, in each [ j − 1, j] (j ∈ Z , coincide with some real polynomial of degree ⩽ m. We study the cardinal spline interpolation problem of constructing an element sϵS m with a( j − λ) = y j ,j ∈ Z where λ∈(0,1] is a translation parameter. Under some natural conditions on λ and m, ter Morsche (in “Spline Functions” ( K. Böhmer, G. Meinardus, and W. Schempp, Eds.), pp. 210–219, Springer-Verlag, Berlin/Heidelberg/New York, 1976) and Schoenberg ( J. Approx. Theory 6 (1972) , 404–420; in “Studies in Spline-Functions and Approximation Theory” ( S. Karlin, Ch. A. Micchelli, A. Pinkus, I. J. Schoenberg, Eds.), pp. 251–276, Academic Press, New York, 1976) have proved that this problem has a unique solution of power growth, provided that the interpolation data are of power growth and that this solution can be given by a series of Lagrangian splines converging locally uniformly. In what follows we prove an analogous result for exponential growth conditions instead of power growth conditions. Moreover, we extend the concept of extremal bases, given by Reimer (in “Approximation Theory III” ( E. Cheney, Ed.), pp. 723–728, Academic Press, New York, 1980), to topological bases of normed spaces with infinite dimension and apply this concept to the subspace of all bounded functions of S m .

On the asymptotic behavior of the bestL ∞-approximation by polynomials

LetEn(f) denote the sup-norm-distance (with respect to the interval [−1, 1]) betweenf and the set of real polynomials of degree not exceedingn. For functions likeex, cosx, etc., the order ofEn(f) asn→∞ is well known. A typical result is $$2^{n - 1} n!E_{n - 1} (e^x ) = 1 + 1/4n + O(n^{ - 2} ).$$ It is shown in this paper that 2n−1n!En−1(ex) possesses a complete asymptotic expansion. This result is contained in the more general result that for a wide class of entire functions (containing, for example, exp(cx), coscx, and the Bessel functionsJk(x)) the quantity $$2^{n - 1} n!E_{n - 1} \left( f \right)/f^{(n)} \left( 0 \right)$$ possesses a complete asymptotic expansion (providedn is always even (resp. always odd) iff is even (resp. odd)).

Extreme points in the family of typically real polynomials

In this paper, we characterize the extreme points in the family of typically real polynomials of degree n or less. Viewed as a subset of the coefficient region which is a subset of n− 1 space, the set of extreme points is the union of two manifolds. Incase n is even, each of the two manifolds is determined by (n−2)/2 parameters that vary in [−1, 1]. If n is odd, then one of the manifolds is determined by(n−l)/2 parameters, and the other by (n −3)/2 parameters. We also determine sharp bounds for some of the coefficients.

Newton’s method and symbolic dynamics

By the use of symbolic dynamics, this note proves a result of B. Barna concerning real polynomials of degree at least 4 and having all distinct simple real roots. Specifically, the set of initial points, for which Newton's method fails to converge to a root of the given polynomial, is homeomorphic to a Cantor set. Also the note shows that the requirement for simple roots may be relaxed, and one still has Barna's result being valid.

Planar Polynomial Foliations

Let $P(x,y)$ and $Q(x,y)$ be two real polynomials of degree $\leqslant n$ with no common real zeros. The solution curves of the vector field $\dot x = P(x,y),\dot y = Q(x,y)$ give a foliation of the plane. The leaf space $\mathcal {L}$ of this foliation may not be a hausdorff space: there may be leaves L, $L’ \in \mathcal {L}$ which cannot be separated by open sets. We show that the number of such leaves is at most 2n and construct an example, for each even $n \geqslant 4$, of a planar polynomial foliation of degree n whose leaf space contains $2n - 4$ such leaves.

Inequalities and Inverse Theorems in Restricted Rational Approximation Theory

The following lemma, part a) due to S. N. Bernstein and part b) due to A. A. Markov, is fundamental to the proofs of many inverse theorems in polynomial approximation theory.LEMMA 1. [2, p. 62 and p. 67] Let ∏n denote the real polynomials of degree at most n. Let p ∈ ∏n, thena)b)From the lemma one can deduce, for example:THEOREM 1. Iƒ there is a sequence of polynomials pn ∈ ∏n and a δ > 0 so that ‖f – pn‖[a, b] ≧ A/nk + δ then f is k times continuously differentiate on (a, b).We shall refer to inequalities, such as those of Lemma 1 that bound the derivative r′ of a rational function of degree n in terms of its supremum norm ‖r‖[a, b] and n, as Bernstein-type inequalities.

Planar polynomial foliations

Let P ( x , y ) P(x,y) and Q ( x , y ) Q(x,y) be two real polynomials of degree ⩽ n \leqslant n with no common real zeros. The solution curves of the vector field x ˙ = P ( x , y ) , y ˙ = Q ( x , y ) \dot x = P(x,y),\dot y = Q(x,y) give a foliation of the plane. The leaf space L \mathcal {L} of this foliation may not be a hausdorff space: there may be leaves L, L ′ ∈ L L’ \in \mathcal {L} which cannot be separated by open sets. We show that the number of such leaves is at most 2n and construct an example, for each even n ⩾ 4 n \geqslant 4 , of a planar polynomial foliation of degree n whose leaf space contains 2 n − 4 2n - 4 such leaves.

Rational approximation to ex and to related functions

According to a well-known result of S. N. Bernstein, e x can be approximated uniformly on [−1, 1] by polynomials of degree ⩽ n with an error of the order [2 n ( n + 1)!] −1. In this note it is shown that the smallest (uniform norm) error in approximating e x by reciprocals of polynomials of degree ⩽ n is also of the order [2 n ( n + 1)!] −1. We denote throughout by P n ( x), Q n ( x) real polynomials of degree ⩽ n. We show, furthermore, that the smallest error in approximating e x by rational functions of the form P n(x) Q n(x) where the coefficients of Q n are ⩾0 is again of that same order.

Numerical solution of linear algebraic equations where the coefficient matrix is a polynomial of a square matrix

To solve the linear algebraic equationP(A)x=y whereP is a real polynomial of degree two, we shall use a stationary iterative method. It is shown that this method converges for all matrices with eigenvalues in a sector in the right complex half plane provided that the zeros ofP are not in the same sector.

20.—Deficiency Indices of Polynomials in Symmetric Differential Expressions, II

SynopsisGiven a symmetric (formally self-adjoint) ordinary linear differential expression L which is regular on the interval [0, ∞) and has C∞ coefficients, we investigate the relationship between the deficiency indices of L and those of p(L), where p(x) is any real polynomial of degree k > 1. Previously we established the following inequalities: (a) For k even, say k = 2r, N+(p(L)), N−(p(L)) ≧ r[N+(L)+N−(L)] and (b) for k odd, say k = 2r+1where N+(M), N−(M) denote the deficiency indices of the symmetric expression M (or of the minimal operator associated with M in the Hilbert space L2(0, ∞)) corresponding to the upper and lower half-planes, respectively. Here we give a necessary and sufficient condition for equality to hold in the above inequalities.

Discrete real time flows with quasi-discrete spectra and algebras generated by expq(t)

These notes (essentially unedited) were sent to W. Parry in 1964. The first two parts are complete and in a letter to Parry at that time Hahn indicated his intention to publish them. Evidently he did not manage to do this. The remainder of these notes represents an attempt to establish a theory of quasidiscrete spectra for discrete one-parameter flows. Hahn indicates the gaps and in a following note Parry clarifies his theory. The first part of these notes presents a characteristic example of a discrete one-parameter flow with quasidiscrete spectrum. Ergodicity, minimality and distality are established. The second part examines the Banach algebra of functions onR generated by {expq(t): q a real polynomial of degree <n+1} and shows that the shift isometries arise from a discrete one-parameter flow on its maximal ideal space Λ n and that ifn is finite this flow is isomorphic to the example examined in the first part.