Taylor Domination, Difference Equations, and Bautin Ideals
Abstract. We compare three approaches to studying the behavior of an analytic function from its Taylor coefficients. The first is “Taylor domination” property for in the complex disk , which is an inequality of the form
The second approach is based on a possibility to generate via recurrence relations. Specifically, we consider linear non-stationary recurrences of the form
with uniformly bounded coefficients.
In the third approach we assume that are polynomials in a finite-dimensional parameter We study “Bautin ideals” generated by in the ring of polynomials in .
These three approaches turn out to be closely related. We present some results and questions in this direction.
Let be an analytic function represented in a disk of radius , centered at the origin by a convergent Taylor series . We assume that the Taylor coefficients are explicitly known, or, at least, can be recovered by a certain explicit procedure. In many cases this is the only analytic information we possess on . A notorious example is the “Poincaré first return mapping” of a non-linear ordinary differential equation (see Section 4.2 below). We would like to investigate the behavior of on the base of what we know, i.e. properties of the sequence . In particular, we would like to bound from above the possible number of the zeroes of . In Section 2 below we provide an explicit connection between the two.
The main goal of this paper is to compare three approaches to the above question: the first is “Taylor domination”, which is a bound on all the Taylor coefficients of through the first few of them. The second is a possibility to generate via Difference Equations, specifically, by linear non-stationary homogeneous recurrence relations of a fixed length, with uniformly bounded coefficients. In the third approach we assume that are polynomials in a finite-dimensional parameter and study “Bautin ideals” generated by in the ring of polynomials in .
The main facts which we present are the following:
1. A sequence can be obtained as a solution of a non-stationary linear homogeneous recurrence relations of a fixed length, with uniformly bounded coefficients, if and only if each its subsequence possesses an appropriate Taylor domination property.
2. A sequence of polynomials in , under some natural assumptions, possesses a uniform in Taylor domination, with the parameters determined through the algebra of the Bautin ideals.
3. If a sequence of polynomials in is produced by an algebraic recurrence relation, its Bautin ideals can be computed explicitly. We discuss briefly the difficulties which arise for differential-algebraic recurrences (like in case of Poincaré mapping of Abel differential equation).
In our discussion we present some new specific results, and many known ones (some of them very recent, some pretty old). We believe that a general picture of interconnections between Taylor domination, recurrence relations, and Bautin ideals, given in this paper is new and may be instructive in further developments.
2 Taylor domination and counting zeroes
“Taylor domination” for an analytic function is an explicit bound of all its Taylor coefficients through the first few of them. This property was classically studied, in particular, in relation with the Bieberbach conjecture, which asserts that for univalent it always holds that . See bieberbach1955analytische (); biernacki1936fonctions (); hayman1994multivalent () and references therein. To give an accurate definition, let us assume the radius of convergence of the Taylor series for is , .
Let a positive finite a natural , and a positive sequence of a subexponential growth be fixed. The function is said to possess an - Taylor domination property if for each we have
If we call this property -Taylor domination.
The parameters of Taylor domination are not defined uniquely. In fact, the following easy result of Bat.Yom.2014 () shows that each nonzero analytic function possesses this property:
[Proposition 1.1, Bat.Yom.2014 ()] If is the radius of convergence of , with , then for each finite and positive , satisfies the -Taylor domination property with being the index of its first nonzero Taylor coefficient, and for .
Consequently, the Taylor domination property becomes really interesting only for those families of analytic functions where we can specify the parameters in an explicit and uniform way. We concentrate on this problem below. Now we recall some well-known features of Taylor domination. Basically, it allows us to compare the behavior of with the behavior of the polynomial . In particular, the number of zeroes of can be easily bounded in this way. In one direction the bound is provided by the classical result of biernacki1936fonctions (). To formulate it, we need the following definition (see hayman1994multivalent () and references therein):
A function regular in a domain is called -valent there, if for any the number of solutions in of the equation does not exceed .
Theorem 2.1 (Biernacki, 1936, biernacki1936fonctions ())
If is -valent in the disk of radius centered at then for each
where is a constant depending only on .
In our notations, Theorem 2.1 claims that a function which is -valent in possesses a - Taylor domination property.
For univalent functions, i.e. for Theorem 2.1 gives for each , while the sharp bound of the Bieberbach conjecture is .
Various forms of inverse results to Theorem 2.1 are known. In particular, an explicit bound for the number of zeroes of possessing Taylor domination can be obtained by combining Proposition 1 and Lemma 2.2 from roytwarf1997bernstein ():
Let the function possess an - Taylor domination property. Then for each , has at most zeros in , where is a function depending only on , and on the sequence , satisfying and for sufficiently small.
We can replace the bound on the number of zeroes of by the bound on its valency, if we exclude in the definition of Taylor domination (or, alternatively, if we consider the derivative instead of ).
3 Taylor domination and recurrence relations
We start with a very general result obtained in roytwarf1997bernstein (). Assume that a sequence of mappings is given, . For any construct a sequence as follows: , and for . We also consider a power series . Of course, any recurrence relation produces such a sequence by iteration.
Assume also that each is a Lipschitzian mapping, satisfying
for any and any , with some given and . This is the case in most of natural examples.
(Theorem 4.1, roytwarf1997bernstein ()) For and as above and for any , the series converges on and possesses there -Taylor domination, with .
Next we restrict ourselves to linear recurrences, and produce more explicit bounds. We consider the class of linear non-stationary homogeneous recurrence relations of a fixed length, with uniformly bounded coefficients:
If for the coefficients have a form , with fixed and with , then recurrence relation (2) is said to be a linear recurrence relation of Poincaré type (see perron1921summengleichungen (); poincare1885equations ()). We denote the class of such recurrences
We would like to write the bounds on in a form
for certain positive constants . So for each we define and to be the pair of constants providing the required bounds on , for which the product is minimal possible. We put
(Theorem 3.1, Bat.Yom.2014 ())
Let be a solution of the recurrence relation
. Put Then the series
converges in the open disk and possesses there
By a proper rescaling, Theorem 3.2 can be easily extended to non-stationary linear recurrences with a subexponential (or exponential) growth of the coefficients . Consequently, generating functions of such recurrences allow for explicit bounds on their valency. On the other hand, a drawback of this approach is that in the case of linear recurrences with constant coefficients (and for Poincaré-type recurrences - see below) the disk where the uniform Taylor domination is guaranteed, is much smaller than the true disk of convergence.
An important feature of Theorem 3.2 is that it allows us to provide an essentially complete characterisation of solutions of recurrence relations through Taylor domination. The following result is new, although it follows closely the lines of Theorem 2.3 of Fri.Yom (). Accordingly, we give only a sketch of the proof, referring the reader to Fri.Yom () for details.
A sequence is a solution of the recurrence relation of length if and only if for each its subsequence possesses a Taylor domination for some positive and .
In one direction the result follows directly from Theorem 3.2. Conversely, if for each the subsequence possesses a Taylor domination, we use the corresponding bound on to construct step by step the coefficients in in such a way that they remain uniformly bounded in . ∎∎
Let us now consider Poincaré-type recurrences. Characterization of their solutions looks a much more challenging problem than the one settled in Theorem 3.3. Still, one can expect deep connections with Taylor domination. One result in this direction (which is a sharpened version of Theorem 3.2 above), was obtained in Bat.Yom.2014 (). For the characteristic polynomial and the characteristic roots of are those of its constant part. We put
(Theorem 5.3, Bat.Yom.2014 ()) Let satisfy a fixed recurrence . Put Let be the minimal of the numbers such that for all we have . We put and
Then possesses -Taylor domination property.
For Poincaré-type recurrences one can ask for Taylor domination in the maximal disk of convergence, which is typically We discuss this problem in the next section.
3.1 Turán’s lemma, and a possibility of its extension
It is well known that the Taylor coefficients of a rational function of degree satisfy a linear recurrence relation with constant coefficients
where are the coefficients of the denominator of . Conversely, for any initial terms the solution sequence of (3) forms a sequence of the Taylor coefficients of a rational function as above. Let be the characteristic roots of (3), i e. the roots of its characteristic equation
Taylor domination property for rational functions is provided by the following theorem, obtained in Bat.Yom.2014 (), which is, essentially, equivalent to the “first Turán lemma” (turan1953neue (); turan1984new (); Ineq (), see also nazarov1994local ()):
Theorem 3.5 provides a uniform Taylor domination for rational functions in their maximal disk of convergence , in the strongest possible sense. Indeed, after rescaling to the unit disk the parameters of (4) depend only on the degree of the function, but not on its specific coefficients.
We consider a direct connection of Turán’s lemma to Taylor domination, provided by Theorem 3.5 as an important and promising fact. Indeed, Turán’s lemma has numerous applications and conncetions, many of them provided already in the first Turan’s book turan1953neue (). It can be considered as a result on exponential polynomials, and in this form it was a starting point for many deep investigations in Harmonic Analysis, Uncertainty Principle, Analytic continuation, Number Theory (see Ineq (); nazarov1994local (); turan1953neue (); turan1984new () and references therein). Recently some applications in Algebraic Sampling were obtained, in particular, estimates of robustness of non-uniform sampling of “spike-train” signals (Sampl (); friedland2011observation ()). One can hope that apparently new connections of Turán’s lemma with Taylor domination, presented in Bat.Yom.2014 () and in the present paper, can be further developed.
A natural open problem, motivated by Theorem 3.5, is a possibility to extend uniform Taylor domination in the maximal disk of convergence , as provided by Theorem 3.5 for rational functions, to wider classes of generating functions of Poincaré type recurrence relations. Indeed, for such functions the radius of convergence of the Taylor series is, essentially, the same as for the constant-coefficients recurrences - it is the inverse of one of the characteristic roots: for some
O.Perron proved in perron1921summengleichungen () that this relation holds for a general recurrence of Poincaré type, but with an additional condition that for all . In pituk2002more () M.Pituk removed this restriction, and proved the following result.
Theorem 3.6 (Pituk’s extension of Perron’s Second Theorem, pituk2002more ())
Let be any solution to a recurrence relation of Poincaré class . Then either for or
where is one of the characteristic roots of .
This result implies the following:
(Theorem 5.2, Bat.Yom.2014 ()) Let be any nonzero solution to a recurrence relation of Poincaré class with initial data , and let be the radius of convergence of the generating function . Then necessarily , and in fact where is some (depending on ) characteristic root of . Consequently, satisfies -Taylor domination with as defined in Proposition 1.
Taylor domination in the maximal disk of convergence provided by Theorem 3.7, is only partially effective. Indeed, the number and the radius are as prescribed by the constant part of the recurrence. However, Proposition 1 only guarantees that the sequence is of subexponential growth but gives no further information on it. We can pose a natural question in this direction. For a sequence of positive numbers tending to zero, consider a subclass of , consisting of with
Problem 1. Do solutions of recurrence relations possess -Taylor domination in the maximal disk of convergence , with depending only on and ? Is this true for specific , in particular, for as it occurs in most of examples (solutions of linear ODE’s, etc.)?
Taking into account well known difficulties in the analysis of Poincaré-type recurrences, this question may be tricky. Presumably, it can be easier for with .
Some initial examples in this direction were provided in yomdin2014Bautin (), via techniques of Bautin ideals. In Section 4 below we indicate an interrelation of the Poincaré-type recurrences with the Bautin ideals techniques. Another possible approach may be via an inequality for consecutive moments of linear combinations of -functions, provided by Theorem 3.3 and Corollary 3.4 of yomdin2010sing.Prony (). This inequality is closely related to Turán’s lemma (and to the “Turán third theorem” of turan1953neue (); turan1984new (); Ineq ()). It was obtained via techniques of finite differences which, presumably, can be extended to Stieltjes transforms of much wider natural classes of functions, like piecewise-algebraic ones.
One can consider some other possible approaches to the “Turán-like” extension of Taylor domination to the full disk of convergence for the Poincaré-type recurrences. First, asymptotic expressions in bodine2004asymptotic (); pituk2002more () may be accurate enough to provide an inequality of the desired form. If this is the case, it remains to get explicit bounds in these asymptotic expressions.
Second, one can use a “dynamical approach” to recurrence relation (2) (see borcea2011parametric (); coppel1971dichotomies (); kloeden2011non (); potzsche2010geometric (); yom.nonaut.dyn () and references therein). We consider (2) as a non-autonomous linear dynamical system . A “non-autonomous diagonalization” of is a sequence of linear changes of variables, bringing this system to its “constant model” , provided by the limit recurrence relation (3).
If we could obtain a non-autonomous diagonalization of with an explicit bound on the size of the linear changes of variables in it, we could get the desired inequality as a pull-back, via , of the Turán inequality for . There are indications that this approach may work in the classes with .
The following partial result in the direction of Problem 1 above was obtained in Bat.Yom.2014 (). It provides Taylor domination in a smaller disk, but with explicit parameters, expressed in a transparent way through the constant part of , and through the size of the perturbations.
(Corollary 5.1, Bat.Yom.2014 ()) Let be a sequence of positive numbers tending to zero. Define as a minimal number such that for we have . Then for each , the solution sequences of possess -Taylor domination, where
3.2 An example: D-finite functions
In this section we briefly summarize results of Bat.Yom.2014 (), concerning a certain class of power series, defined by the Stieltjes integral transforms
where belongs to the class of the so-called piecewise D-finite functions bat2008 (), which are solutions of linear ODEs with polynomial coefficients, possessing a finite number of discontinuities of the first kind.
Using the expansion for , we obtain the following useful representation of through the moments of :
Obtaining uniform Taylor domination for where belongs to particular subclasses of (in particular, being piecewise algebraic), is an important problem with direct applications in Qualitative Theory of ODEs (see Bli.Bri.Yom (); briskin2010center () and references therein).
A real-valued bounded integrable function is said to belong to the class if it has discontinuities (not including the endpoints ) of the first kind, and between the discontinuities it satisfies a linear homogeneous ODE with polynomial coefficients , where
Let , with as above. Denote the discontinuities of by . In what follows, we shall use some additional notation. Denote for each Let .
Our approach in Bat.Yom.2014 () is based on the following result:
Theorem 3.9 (bat2008 ())
Let . Then the moments satisfy the recurrence relation
where are polynomials in expressed through the coefficients of , while in the right hand side are expressed through the values of the coefficients of , and through the jumps of , at the points .
The vector function is said to satisfy a linear system of Poincaré type, if
where is a constant matrix and is a matrix function satisfying .
Put . Now define the vector function as
The vector function satisfies a linear system of the form (7). This system is of Poincaré type, if and only if
This last condition is equivalent to the operator having at most a regular singularity at . The set of the eigenvalues of the matrix is precisely the union of the roots of (i.e. the singular points of the operator ) and the jump points .
Theorem 3.10 (pituk2002more ())
Let the vector satisfy the perturbed linear system of Poincaré type (7). Then either for or
exists and is equal to the modulus of one of the eigenvalues of the matrix .
Next main problem is: how many first moments can vanish for a nonzero ?. in Taylor domination cannot be smaller than this number. In batbinZeros () we study this question, proving the following result.
Theorem 3.11 (batbinZeros ())
Let the operator be of Fuchsian type (i.e. having only regular singular points, possibly including ). In particular, satisfies the condition (8). Let .
If there is at least one discontinuity point of at which the operator is nonsingular, i.e. , then vanishing of the first moments of implies .
Otherwise, let denote the largest positive integer characteristic exponent of at the point . In fact, the indicial equation of at is . Then the vanishing of the first moments of implies .
With these theorems in place, the following result is obtained in Bat.Yom.2014 ():
4 Bautin Ideals
In this section we consider families
where . We assume that each is a polynomial in . It was a remarkable discovery of N. Bautin (Bau1 (); Bau2 ()) that in this situation the behavior of the Taylor coefficients of , and consequently, of its zeroes, can be understood in terms of the ideals generated by the subsequent Taylor coefficients in the polynomial ring . Explicit computation of in specific examples may be very difficult. In particular, Bautin himself computed for the Poincaré first return mapping of the plane polynomial vector field of degree , producing in this way one of the strongest achievements in the Hilbert 16th problem up to this day: at most three limit cycles can bifurcate from an isolated equilibrium point in vector field of degree .
Bautin’s approach can be extended to a wide classes of families of the form (9). In particular, the following class was initially defined and investigated in Bri.Yom (); Fra.Yom (); roytwarf1997bernstein (), and further studied in Yom.Baut (); yomdin2014Bautin ():
Let . is called an -series if the following condition is satisfied:
for some positive .
The ideal generated by all the subsequent Taylor coefficients in the polynomial ring is called the Bautin ideal of , and the minimal such that is called the Bautin index of .
Such a finite exists by the Noetherian property of the ring .
The following result of Fra.Yom () connects the algebra of the Bautin ideal of with Taylor domination for this series:
Let be an -series, and let and be the Bautin ideal and the Bautin index of . Then possesses a -Taylor domination for some positive depending on , and on the basis of the ideal .
The idea of the proof is very simple: if for the -series the first Taylor coefficients generate the ideal , then for any we have
with - certain polynomials in . The classical Hironaka’s division theorem provides a bound on the degree and the size of , and hence we get an explicit bound on through for each . ∎∎
Also here to get an explicit and uniform in Taylor domination in the full disk of convergence is a difficult problem. Some general additional conditions on -series, providing such result, were given in Yom.Baut (). These conditions are technically rather involved. A simpler special case was treated in yomdin2014Bautin ().
4.1 Recurrence relations and Bautin ideals
A rather detailed investigation of -series produced by recurrence relations was provided in Bri.Yom (). It was done in several situations, including algebraic functions, linear and nonlinear differential equations with polynomial coefficients, and the Poincaré first return mapping for the Abel differential equation. In all these cases, except the last one, it was shown that the resulting series are -ones, and their Bautin ideal was computed. In the present paper we generalize the result of Bri.Yom (), showing how to compute the Bautin ideal for general non-stationary polynomial recurrences. Let, as above, , and put Consider a sequence of polynomials
are, in their turn, polynomials in of degrees . Here , and are multi-indices, and Finally, are complex constants.
A recurrence of the form
is called a non-stationary polynomial recurrence relation of length .
The following result is purely algebraic, so we put no restrictions on the degrees and the size of the polynomials .
Let polynomials in be given, and let be produced by recurrence (10). Then the Bautin ideal of the formal series is generated by
Equation (10) applied with shows that belongs to the ideal generated by Applying this equation step by step, we see that all belong to . Therefore . ∎∎
We expect that if we assume the degrees and , as well as the coefficients to be uniformly bounded, then is, in fact, an -series. We expect that this fact can be shown by the methods of Bri.Yom (), properly extended to non-stationary polynomial recurrence relation. Let us consider an example in this direction.
We consider linear recurrence
Let polynomials of degrees in be given, and let be produced by recurrence (11), with uniformly bounded in . Then the formal series is, in fact, an series. In particular, the degree of is at most , and the Bautin ideal is generated by
Equation (11) shows that the degree in of is by one higher than the maximum of the degrees of . Applying induction we see that the degree of is at most . The last statement of Proposition 3 follows directly from Theorem 4.2. Now, write
Then equation (11) shows that satisfy a matrix recurrence relation, which in coordinates takes a form
Here for we define as
So in the right hand side of (12) appear all the with between and , and smaller than by one in exactly one coordinate.
Applying to the matrix recurrence (12) straightforward estimates, we obtain exponential in upper bounds on the coefficients , so is indeed an series. ∎∎
4.2 Poincaré coefficients of Abel equation
Consider Abel differential equation
with polynomial coefficients on the interval . A solution of (13) is called “closed” if . The Smale-Pugh problem, which is a version of the (second part of) Hilbert’s 16-th problem, is to bound the number of isolated closed solutions of (13) in terms of the degrees of and .
This problem can be naturally expressed in terms of the Poincaré “first return” mapping along . Let denote the solution of (13) satisfying . The Poincaré mapping associates to each initial value at the value at of the solution analytically continued along . Closed solutions correspond to the fixed points of . So the problem is reduced to bounding the number of the fixed points of , or of zeroes of . Historically, one of the most successful directions in the study of the Poincaré mapping was the direction initiated by Bautin in Bau1 (); Bau2 (): to derive the analytic properties of , in particular, the number of its fixed points, from the structure of its Taylor coefficients.
It is well known that for small is given by a convergent power series
The “Poincaré coefficients” of the Poincaré mapping from to satisfy the following differential-recurrence relation (see, for example, briskin2010center ()):
This recurrence is apparently not of the form considered above, i.e. it is not “polynomial recurrence”. Still, it is easy to see from (15) that the Poincaré coefficients are polynomials with rational coefficients in the parameters of the problem (i.e. in the coefficients of and ). They can be explicitly computed for as large indices as necessary. However, because of the derivative in the left hand side, (15) does not preserve ideals. So Theorem 4.2 is not applicable to it, and computing Bautin ideals for the Poincaré mapping is in general a very difficult problem. Also Taylor domination for is not well understood, besides some special examples. The only general result concerning the Bautin ideals of we are aware of was obtained in Bli.Bri.Yom () using an approximation of the “Poincaré coefficients” with certain moment-like expressions of the form . We would like to pose the investigation of the recurrence relation (15) in the lines of the present paper as an important open problem.
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