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Abstract
I give a short and completely elementary proof of Takagi’s 1921 theorem on the zeros of a composite polynomial .
Many theorems in the analytic theory of polynomials [Citation2, Citation8, Citation10, Citation11] are concerned with locating the zeros of composite polynomials. More specifically, let f and g be polynomials (with complex coefficients) and let h be a polynomial formed in some way from f and g; under the assumption that the zeros of f (respectively, g) lie in a subset S (respectively, T) of the complex plane, we wish to deduce that the zeros of h lie in some subset U. The theorems are distinguished by the nature of the operation defining h, and the nature of the subsets S, T, U under consideration.
Here we shall be concerned with differential composition: , or
for short. In detail, if
and
, then
; and D denotes the differentiation operator, i.e.,
. The following important result was found by Takagi [Citation13] in 1921, subsuming many earlier results:Footnote1
Theorem 1
(Takagi). Let f and g be polynomials with complex coefficients, with and
. Let f have an r-fold zero at the origin (
), and let the remaining zeros (with multiplicity) be
. Let K be the convex hull of the zeros of g. Then either
is identically zero, or its zeros lie in the set
.
Here we have used the notations and
.
Takagi’s proof was based on Grace’s apolarity theorem [Citation3], a fundamental but somewhat enigmatic result in the analytic theory of polynomials.Footnote2 This proof is also given in the books of Marden [Citation8, Section 18], Obrechkoff [Citation10, pp. 135–136], and Rahman and Schmeisser [Citation11, Sections 5.3 and 5.4]. Here I give a short and completely elementary proof of Takagi’s theorem.
The key step—as Takagi [Citation13] observed—is to understand the case of a degree-1 polynomial :
Proposition 2
(Takagi). Let g be a polynomial of degree n, and let K be the convex hull of the zeros of g. Let , and define
. Then either h is identically zero, or all the zeros of h are contained in K if α = 0, and in
if
.
The case α = 0 is the celebrated theorem of Gauss and Lucas [Citation8, Section 6], [Citation10, Chapter V], and [Citation11, Section 2.1], which is the starting point of the modern analytic theory of polynomials. My proof for general α will be modeled on Cesàro’s [Citation1] 1885 proof of the Gauss–Lucas theorem [Citation11, pp. 72–73], with a slight twist to handle the case .
Proof of Proposition 2.
Clearly, h is identically zero if and only if either (a) or (b) g is a nonzero constant and α = 0. Moreover, if g is a nonzero constant and
, then the zero set of h is empty. So we can assume that
.
Let be the zeros of g (with multiplicity), so that
with
. If
, then
, and we can consider
If this equals zero, then by taking complex conjugates we obtainwhich can be rewritten as
where
Then and
, so
; and of course
. Moreover, by the Schwarz inequality we have
so . This implies that
and hence that
. ■
We can now handle polynomials f of arbitrary degree by iterating Proposition 2:
Proof of Theorem 1.
From it is easy to see that
. We first apply Dr to g, yielding a polynomial of degree n – r whose zeros also lie in K (by the Gauss–Lucas theorem); then we repeatedly apply (in any order) the factors
, using Proposition 2. ■
Remark.
When α = 0, the zeros of lie in K; so one might expect that when α is small, the zeros of
should lie near K. But when α is small and nonzero, the set
arising in Proposition 2 is in fact very large. What is going on here?
Here is the answer: Suppose that . When α = 0, the polynomial
has degree n – 1; but when
, the polynomial
has degree n. So, in order to make a proper comparison of their zeros, we should consider the polynomial
corresponding to the case α = 0 as also having a zero “at infinity.” This zero then moves to a value of order
when α is small and nonzero.
This behavior is easily seen by considering the example of a quadratic polynomial . Then the zeros of
are
So there really is a zero of order , as Takagi’s theorem recognizes.
In the context of Proposition 2, one expects that has one zero of order
and n – 1 zeros near K (within a distance of order α). More generally, in the context of Theorem 1, one would expect that h has m – r zeros of order
, with the remaining zeros near K. It is a very interesting problem — and one that is open, as far as I know — to find strengthenings of Takagi’s theorem that exhibit these properties. There is an old result that goes in this direction [Citation8, Corollary 18.1], [Citation11, Corollary 5.4.1(ii)], but it is based on a disc D containing the zeros of g, which might in general be much larger than the convex hull K of the zeros.
Postscript. A few days after finding this proof of Proposition 2, I discovered that an essentially identical argument is buried in a 1961 paper of Shisha and Walsh [Citation12, pp. 127–128 and 147–148] on the zeros of infrapolynomials. I was led to the Shisha–Walsh paper by a brief citation in Marden’s book [Citation8, pp. 87–88, Exercise 11]. So the proof given here is not new; but it deserves to be better known.
Acknowledgment
This research was supported in part by U.K. Engineering and Physical Sciences Research Council grant EP/N025636/1.
Additional information
Funding
Notes
1 See Honda [Citation4], Iyanaga [Citation5, Citation6], Kaplan [Citation7], and Miyake [Citation9] for biographies of Teiji Takagi (inlinefigure, Takagi Teiji, 1875–1960). Takagi’s papers published in languages other than Japanese (namely, English, German, and French) have been collected in [Citation14].
2 For discussion of Grace’s apolarity theorem and its equivalents—notably Walsh’s coincidence theorem and the Schur–Szegő composition theorem—see Marden [Citation8, Chapter IV], Obrechkoff [Citation10, Chapter VII], and especially Rahman and Schmeisser [Citation11, Chapter 3].
References
- Cesàro, E. (1885). Solution de la question 1338. Nouvelles Annales de Mathématiques (3e série). 4: 328–330. www.numdam.org/article/NAM_1885_3_4__328_0.pdf
- Dieudonné, J. (1938). La Théorie Analytique des Polynômes d’une Variable (à Coefficients Quelconques). Mémorial des Sciences Mathématiques, fascicule 93. Paris: Gauthier-Villars. www.numdam.org/issue/MSM_1938__93__1_0.pdf
- Grace, J. H. (1902). The zeros of a polynomial. Proc. Cambridge Philos. Soc. 11: 352–357.
- Honda, K. (1975). Teiji Takagi: A biography — on the 100th anniversary of his birth. Comment. Math. Univ. St. Paul. 24(2): 141–167. DOI: https://doi.org/10.14992/00010342.
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- Kaplan, P. (1997). Takagi Teiji et la découverte de la théorie du corps de classes. Ebisu — Études Japonaises. 16: 5–11. www.persee.fr/doc/ebisu_1340-3656_1997_num_16_1_973
- Marden, M. (1966). Geometry of Polynomials, 2nd ed. Providence, RI: American Mathematical Society. (First edition 1949.)
- Miyake, K. (2007). Teiji Takagi, founder of the Japanese school of modern mathematics. Japanese J. Math. 2(1): 151–164. DOI: https://doi.org/10.1007/s11537-007-0649-8.
- Obrechkoff, N. (2003). Zeros of Polynomials. Sofia: Marin Drinov Academic Publishing House. (Originally published in Bulgarian: Obreškov, N. (1963). Nuli na Polinomite. Sofia: Izdat. Bǔlgar. Akad. Nauk.)
- Rahman, Q. I., Schmeisser, G. (2002). Analytic Theory of Polynomials. Oxford: Clarendon Press.
- Shisha, O., Walsh, J. L. (1961). The zeros of infrapolynomials with some prescribed coefficients. J. Analyse Math. 9: 111–160. DOI: https://doi.org/10.1007/BF02795341.
- Takagi, T. (1921). Note on the algebraic equations. Proc. Phys.-Math. Soc. Japan. 3(11): 175–179. DOI: https://doi.org/10.11429/ppmsj1919.3.11_175. (Reprinted in [14, pp. 175–178].)
- Takagi, T. (1990). Collected Papers, 2nd ed. Edited and with a preface by S. Iyanaga, K. Iwasawa, K. Kodaira, and K. Yosida. Tokyo: Springer-Verlag. (Reprinted by Springer-Verlag, Heidelberg, 2014.)