Royal Society Publishing

The number of zeros of a sum of fractional powers

G.J.O Jameson

Abstract

We consider functions of the form Embedded Image, where Embedded Image. A version of Descartes's rule of signs applies. Further, if Embedded Image and Embedded Image, then the number of zeros of f is bounded by the number of sign changes of Embedded Image. The estimate is reduced by 1 for each relation of the form Embedded Image.

Keywords:

1. Introduction

We consider functions of the formEmbedded Image(1.1)with p real, the Embedded Image non-zero andEmbedded Image(1.2)We call these functions sums of fractional powers, though in fact we do not exclude integer values of p. The number of terms, n, is the length of f. Our objective is to give bounds for the number of zeros of such functions, counted with their orders. Denote this number by Embedded Image. Of course, we have to exclude the case when Embedded Image is identically zero; this can only happen if p is one of the integers Embedded Image.

A special case of particular interest is when each Embedded Image is either 1 or −1, with equally many of each occurring. It is then natural to use notation likeEmbedded Imagewhere Embedded Image. We will call this type bipartite.

One context in which this problem arises is in relation to the method of Bombieri & Iwaniec for exponential sums, as developed in Watt (1989) and Huxley (1996), ch. 11. For rational p, say Embedded Image, Watt (lemma 4.2) gives the bound Embedded Image, by moving to an expression involving only integer powers. When Embedded Image, this becomes Embedded Image. The case really wanted is f bipartite of length 8, for which Watt's bound is 64 (though Huxley uses the rather more generous estimate Embedded Image).

A more fruitful approach was initiated by Laguerre as long ago as 1883, taking Descartes's rule of signs as the starting point. For ordinary real polynomials, Descartes's rule states that the number of positive zeros is no greater than the number of sign changes of the coefficients. By a method based on Rolle's theorem, Laguerre extended the rule to generalized polynomials Embedded Image, and Dirichlet polynomials Embedded Image. We show that a similar result applies to functions of type (1.1), without any requirement that p be rational. In particular, it follows that Embedded Image.

For Dirichlet polynomials, Laguerre formulated a powerful variant in which Embedded Image is replaced by the sequence of partial sums Embedded Image. He also obtained a result of this kind for functions of our type, but only for negative p (Laguerre 1898, p. 41). We show that this result extends to the case p>0, subject to the condition Embedded Image; a completely different method seems to be needed. For the bipartite type of length Embedded Image, this implies that Embedded Image.

We then establish an extension of this theorem that has no counterpart for Dirichlet polynomials: the bound for Embedded Image is reduced by one for each relation of the form Embedded Image satisfied by the coefficients. Exactly this assumption (for r=1, 2) is in force for the bipartite functions of length 8 considered by Huxley & Watt, so in fact these functions have at most one zero, a distinct improvement on the estimate 64 (or 768)! This at least leads to some simplification, and better estimates for constants, in the ensuing results in their study.

Laguerre's work in this area does not seem to have been accorded much attention in recent literature. It is reproduced in exercise form, and partly with new methods, in Pólya & Szegö (1964) (Part V, ch. 1). The forthcoming article Jameson (in press) is an expository account with full proofs.

2. Preliminaries

We consider the function f defined by (1.1) on Embedded Image, with the Embedded Image all non-zero. We exclude the trivial case p=0. Note that if p is a positive integer, then f is an ordinary polynomial of degree at most p.

One may as well assume that Embedded Image, since this is effected by the substitution Embedded Image. Effectively, this is just maximizing the domain of f.

We must clarify the possibility of f being identically zero, as in the trivial example Embedded Image. This is easily resolved in the case where p is not a positive integer.

If f is defined by (1.1) and p is not a positive integer, then Embedded Image is not identically zero.

It is enough to show that some derivative of f is not identically zero. Now Embedded Image is a non-zero multiple of Embedded Image. For a fixed x, once r is large enough (so that pr is large enough negative), the term Embedded Image dominates the others, so Embedded Image. ▪

Now consider the case where p is a positive integer. With f as in (1.1), consider the Dirichlet polynomialEmbedded Image(2.1)By the binomial theorem,Embedded Image(2.2)from which it is clear that f is not identically zero, provided that Embedded Image for some r with Embedded Image.

Suppose that f is defined by (1.1) and p is a positive integer. Then f may be identically zero if Embedded Image, but not if Embedded Image.

If Embedded Image, then the vectors Embedded Image Embedded Image are linearly dependent, so there exist Embedded Image, not all 0, such that if G is defined by (2.1), then Embedded Image, for Embedded Image.

Now suppose that Embedded Image, and that distinct Embedded Image and non-zero Embedded Image are given. Then (as is well known) the vectors Embedded Image Embedded Image are linearly independent, so Embedded Image for some Embedded Image. ▪

Hence if p is not one of the integers Embedded Image, then the representation of a function in the form (1.1) is unique.

3. Extension of Descartes's rule of signs

For a function possessing all derivatives, we write Embedded Image for the number of zeros of f in an interval I, counted with their orders. We shorten this to Embedded Image when I is the whole domain Embedded Image of f. Recall that Rolle's theorem implies that Embedded Image.

Denote by Embedded Image the number of sign changes of the sequence Embedded Image, in other words, the number of terms that have the opposite sign to the previous non-zero term. Clearly, if Embedded Image has length n, then Embedded Image.

For a Dirichlet polynomial Embedded Image Embedded Image, with Embedded Image, Laguerre's extension of Descartes's rule of signs states that Embedded Image. The substitutions Embedded Image and Embedded Image transform Embedded Image into the generalized polynomial Embedded Image. In either form, the result applies equally to infinite series (within their interval of convergence) whose coefficients have only finitely many sign changes. The following analogue of Descartes's rule applies to sums of fractional powers.

Suppose that f is defined by (1.1) and (1.2), and is not identically zero. Then Embedded Image.

The proof is by induction on the number of sign changes. If there are no sign changes, then all the Embedded Image have the same sign (say Embedded Image), so Embedded Image for all x>0 and f has no zeros. Assume that the statement is true when there are m sign changes, and suppose that Embedded Image. Let the last sign change occur at the term j=k, so that Embedded Image has the opposite sign to Embedded Image. Choose a, such that Embedded Image. Then f has the same zeros (with the same orders) as Embedded Image, whereEmbedded ImageHenceEmbedded Imageso thatEmbedded ImageNow Embedded Image has the same sign for Embedded Image and j=k. Otherwise, it has the same sign changes as Embedded Image, so it has m sign changes altogether. If Embedded Image is not identically zero, then, by the induction hypothesis, it has at most m zeros. By Rolle's theorem, Embedded Image. If Embedded Image is identically zero, then Embedded Image for a non-zero constant K, so Embedded Image. ▪

Suppose that f is defined by (1.1) and (1.2), and is not constant. Let Embedded Image. Then any non-zero value is assumed by f at most Embedded Image times.

Since Embedded Image and is not identically zero, we have Embedded Image. The statement follows, by Rolle's theorem again. ▪

Corollary 3.2 clearly also holds for the function Embedded Image. However, such functions may also take the value zero Embedded Image (not m) times; for example, the single term Embedded Image is zero when Embedded Image.

Clearly, theorem 3.1 implies that Embedded Image, where n is the length of f. In the usual way, there is an algebraic restatement of this fact.

Let Embedded Image Embedded Image be distinct positive numbers and Embedded Image Embedded Image distinct non-negative numbers. Let p be a real number other than Embedded Image. Then the matrix Embedded Image is non-singular.

However, as the remarks in §2 show, if p had one of the excluded values, then the matrix in corollary 3.3 would be singular for all choices of Embedded Image.

For polynomials, or Dirichlet polynomials, Descartes's rule incorporates the further feature that the difference between Embedded Image and Embedded Image is necessarily even. It is easily seen that this statement does not transfer to functions of our type. For example, Embedded Image has one sign change, but no zeros.

4. Bounds in terms of the sequence (Embedded Image): Laguerre's method

Write Embedded Image. We will present some generalizations of theorem 3.1, in which Embedded Image is replaced by Embedded Image.

First, some elementary facts about Embedded Image. It is not greater than Embedded Image, because each time Embedded Image has a new sign, the corresponding term Embedded Image must have the same sign as Embedded Image. It is possible to have Embedded Image while Embedded Image. In the case where Embedded Image, we have Embedded Image, while Embedded Image, from which it follows that Embedded Image must differ from Embedded Image by an odd integer; in particular, it is not greater than Embedded Image.

Given the condition Embedded Image (but not otherwise!), it makes no difference if the original Embedded Image are listed in reverse order: in fact, if Embedded Image, then Embedded Image, henceEmbedded Imagewhich clearly has the same number of sign changes as Embedded Image.

If Embedded Image is bipartite of length Embedded Image, then Embedded Image.

Suppose that Embedded Image has a sign change at j=k. Since each Embedded Image is 1 or −1, this means that Embedded Image, and the next sign change cannot occur before Embedded Image. The first sign change cannot occur until j=3, so the total number is at most Embedded Image. This number occurs when Embedded Image consists of pairs Embedded Image alternating with Embedded Image. ▪

Laguerre's second theorem for Dirichlet polynomials is as follows. Actually, his reasoning (p. 9) depends on a limiting process which seems to the present author to need further explanation, and the proof of Pólya & Szegö (1925) may have been the first fully satisfactory one. Only version (i) below was stated by these writers, but version (ii) is readily obtained by the same proof.

Let Embedded Image for Embedded Image, where Embedded Image. Then:

  1. Embedded Image,

  2. if Embedded Image, then Embedded Image.

Laguerre (1898), (p. 40–41) derived a corresponding result for functions of our type (1.1) with p<0. His method applies equally to positive integer values of p, and extends naturally to include (ii) below (which he did not state). We outline it here, both because it is short and elegant, and because it serves as a pleasantly simple proof of our main theorem for these values of p.

Suppose that p is a positive integer and that f is defined by (1.1) and is not identically zero. Let Embedded Image, and let Embedded Image. Then:

  1. Embedded Image,

  2. if Embedded Image for some Embedded Image, then Embedded Image.

Under the condition in (ii), we know from proposition 4.2 that Embedded Image.

Recall from (2.2) thatEmbedded ImageBy Descartes's rule of signs for ordinary polynomials,Embedded ImageBy the intermediate value theorem, this is not greater than Embedded Image, which, by proposition 4.2, is not greater than m.

Under the condition in (ii), we haveEmbedded Image ▪

For the case p<0, it is more convenient to express Embedded Image as a combination of terms Embedded Image instead of Embedded Image. This means that the coefficients Embedded Image appearing correspond to the usual Embedded Image's taken in the opposite order; it must be remembered that unless Embedded Image, Embedded Image is not the same as Embedded Image.

Let p<0 andEmbedded Imagewhere Embedded Image. Let Embedded Image, and let Embedded Image. Then:

  1. Embedded Image,

  2. if Embedded Image for some k>0, then Embedded Image.

By an obvious translation, we may assume that Embedded Image. We have Embedded Image for Embedded Image, where Embedded Image for all r. Now Embedded Image, where, for Embedded Image,Embedded Imagewhere Embedded Image. So Embedded Image is the same as the number of zeros of Embedded Image for Embedded Image. By Descartes's rule for power series, this number does not exceed Embedded Image. The proof of both statements now continues as in theorem 4.3. ▪

If Embedded Image, then Rolle's theorem gives Embedded Image. In fact, the method can be modified to give a more precise statement for this case, as follows: Embedded Image, where Embedded Image for Embedded Image. We deduce that Embedded Image, which in turn is no greater than Embedded Image (so Embedded Image unless Embedded Image and Embedded Image are non-zero with the same sign).

5. Bounds in terms of (Embedded Image): the general case

We show that both parts of theorem 4.3 extend to any p, under the extra condition Embedded Image. Note that Embedded Image. The proof of the first part is an adaptation of the method of Pólya & Szegö for proposition 4.2 (Part V, exercises 80, 83).

Let f be defined by (1.1) and (1.2), with Embedded Image. Abel summation givesEmbedded Image(5.1)Hence if Embedded Image for all j, then f has no zeros: in fact, if p>0, then Embedded Image for all x, and if p<0, then Embedded Image for all x.

We rewrite (5.1) as an integral:Embedded Image(5.2)where Embedded Image. Rewrite this again asEmbedded Image(5.3)where Embedded Image for Embedded Image Embedded Image and Embedded Image for other t.

Of course, the function ϕ has sign changes exactly corresponding to those of Embedded Image. Without any attempt at maximum generality, we now formulate a result analogous to theorem 3.1 for functions defined by an integral in this way. Suppose (with a slight change of notation) that ϕ is a function on Embedded Image, such that

  1. there exist points Embedded Image, such that on each open interval Embedded ImageEmbedded Image, ϕ is bounded and continuous and either strictly positive, strictly negative or zero.

We count the point Embedded Image as a sign change of ϕ if it has opposite signs on Embedded Image and the last earlier interval where it was not zero.

Suppose that ϕ, not identically zero, satisfies (A) and has m sign changes in Embedded Image. Let Embedded Image andEmbedded ImageIf g is not identically zero, then Embedded Image.

Induction on m, copying the proof of theorem 3.1. If m=0, then either Embedded Image for all t or Embedded Image for all t: assume the first. Condition (A) now ensures that Embedded Image for all x, so Embedded Image.

Now assume that the theorem is correct for a certain value m and that ϕ has Embedded Image sign changes. Let one of them be at c, and letEmbedded ImageThen, by differentiation under the integral sign,Embedded ImagesoEmbedded ImageNow Embedded Image satisfies condition (A) and has the same sign changes as Embedded Image, except that it does not have one at c. Hence it has m sign changes, and, by the induction hypothesis, Embedded Image (unless Embedded Image is identically zero). Rolle's theorem gives the required statement Embedded Image. ▪

By (5.3), we deduce immediately.

Suppose that f, not identically zero, is defined by (1.1) and (1.2), with Embedded Image. Then Embedded Image.

When p<0, this only reproduces theorem 4.4(i) with the extra condition Embedded Image, but our method can be modified to dispense with this condition. For this purpose, assume that p<0 and that f is given by (1.1), but with the Embedded Image in ascending order: Embedded Image. In (5.1), we can writeEmbedded Imageso (5.2) is modified toEmbedded Imagewhere Embedded Image for Embedded Image and Embedded Image. The proof then continues as before.

Of course, this modification is not possible when p>0. The following example shows that without the condition Embedded Image, Embedded Image can be greater than both Embedded Image and Embedded Image.

Let

Embedded Image

Then Embedded Image. One finds that Embedded Image, Embedded Image and Embedded Image, so f has two zeros.

We now set out to prove our main theorem, which extends proposition 5.2 to a version incorporating the second statement in theorems 4.3 and 4.4. With Embedded Image expressed as in (5.3) and Embedded Image, we have

Embedded Imageso the condition Embedded Image equates to Embedded Image.

The proof is by another induction process like the one in lemma 5.1, but this time on k, keeping mk fixed. We use the following variant of Rolle's theorem for functions that tend to 0 as Embedded Image.

Suppose that f is a function on Embedded Image, possessing all derivatives and having finitely many zeros. Suppose also that Embedded Image as Embedded Image. Then Embedded Image.

Let the last zero of f occur at Embedded Image, and assume that Embedded Image for Embedded Image. There is a point Embedded Image where f attains its greatest value on Embedded Image, and, clearly, Embedded Image. Then Embedded Image has at least Embedded Image zeros in Embedded Image, hence at least Embedded Image zeros altogether. ▪

Reverting to the notation of lemma 5.1, we have to prove:

Suppose that ϕ satisfies (A) and has m sign changes in Embedded Image, and alsoEmbedded Image(5.4)Let Embedded Image andEmbedded ImageIf g is not identically zero, then Embedded Image.

Fix Embedded Image and consider pairs Embedded Image with Embedded Image. The proof is then by induction on k. The case k=0 (so that condition (5.4) is empty) is lemma 5.1. Assume, then, that the result is correct for Embedded Image (where Embedded Image), and that the conditions are as in the statement. Follow the proof of lemma 5.1. NowEmbedded Imageuniformly for Embedded Image. Hence the same is true with both sides multiplied by the bounded function Embedded Image. Since Embedded Image, it follows that Embedded Image as Embedded Image. As before, we haveEmbedded Imagewhere Embedded Image, with Embedded Image sign changes. Also,Embedded Imagefor Embedded Image (if k=1, there is no such statement, and none is needed). By the induction hypothesis, Embedded Image. By lemma 5.4, Embedded Image. (Again, this still holds if Embedded Image is identically zero.) ▪

So we have completed the proof of our main theorem.

Suppose that f, not identically zero, is defined by (1.1) and (1.2). Let Embedded Image. Write Embedded Image, and suppose thatEmbedded Imagefor some Embedded Image. Then Embedded Image and Embedded Image.

Under the same conditions, f attains any given value at most Embedded Image times.

Theorem 5.6 applies equally to Embedded Image (with Embedded Image replaced by Embedded Image), and the statement follows, by Rolle's theorem. ▪

LetEmbedded Image

One checks easily that m=2 and Embedded Image. So f has no positive zeros (except for the cases Embedded Image, when it is identically zero).

The bipartite case. Let Embedded Image. By lemma 4.1, Embedded Image, so if Embedded Image, then f has no zeros, and if Embedded Image, then f has at most one zero. The condition Embedded Image equates to Embedded Image. This is exactly the situation considered by Huxley & Watt (specifically with Embedded Image and k=2).

The case k=m in theorem 5.6 can be deduced directly from proposition 4.2, as follows. Fix x and define (for any q) Embedded Image. By the binomial theorem, the assumption Embedded Image implies that H has zeros at Embedded Image. By proposition 4.2, it has no other zeros, and these zeros are simple, so the sign of Embedded Image alternates on the intervals between them (with Embedded Image for large q, if Embedded Image). Another choice of x will give a new H, but still with signs on these intervals determined in the same way. So, for the given p (assumed not to be one of Embedded Image), Embedded Image (and hence Embedded Image) has the same sign for all choices of x.

One can extend this argument to the case Embedded Image, using the fact that at two successive zeros of f (if they exist), Embedded Image has opposite signs.

Questions about the case Embedded Image. We have seen that proposition 5.2, as stated, does not hold when p>0 and Embedded Image. However, a certain amount can be said about this case. The Abel summation expression shows that if Embedded Image, then Embedded Image. A proof along the lines of the previous note shows that if p>1 and Embedded Image, then Embedded Image. We leave it as an open problem whether these statements can be generalized. One might also ask whether the inequality Embedded Image holds for all p>0.

Acknowledgments

I am grateful to Peter Walker for directing me to the relevant literature, and to the referees for several useful suggestions.

Footnotes

    • Received September 22, 2005.
    • Accepted December 15, 2005.

References

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