×

Integral geometry and real zeros of Thue-Morse polynomials. (English) Zbl 0976.52005

Summary: 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 \(\frac 2\pi \log n + O(1)\) real zeros (M. Kac’s theorem).
This result leads us to the following problem: given a real sequence \((\alpha_k)_{k\in\mathbb{N}}\), study the average \[ \tfrac 1N\sum^{N-1}_{n=0}\rho(f_n), \] where \(\rho(f_n)\) is the number of real zeros of \(f_n(X)=\alpha_0+\alpha_1X+\cdots+\alpha_nX^n\). We give theoretical results for the Thue-Morse polynomials and numerical evidence for other polynomials.

MSC:

52A22 Random convex sets and integral geometry (aspects of convex geometry)
53C65 Integral geometry
11B85 Automata sequences
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] Dekking F. M., J. Reine Angcw. Math. 329 pp 143– (1981)
[2] Doche C., ”On the real roots of generalized Thue–Morse polynomials” (1999) · Zbl 1020.11017
[3] Edelman A., Bull. Amer. Math. Soc. (N.S.) 32 (1) pp 1– (1995) · Zbl 0820.34038
[4] Erdöos P., Proc. London Math. Soc. (3) 6 pp 139– (1956) · Zbl 0070.01702
[5] Favard J., C. R. Acad. Sci. Paris 194 pp 344– (1932)
[6] Kac M., Bull. Amer. Math. Soc. 49 pp 314– (1943) · Zbl 0060.28602
[7] Kac M., Proc. London Math. Soc. (2) 50 pp 390– (1949) · Zbl 0033.14702
[8] Kac M., Probability and related topics in physical sciences (1959) · Zbl 0087.33003
[9] Klain D. A., Introduction to geometric probability (1997) · Zbl 0896.60004
[10] Laurent-Gengoux C., Jour. Math. des éeléeves ENS.
[11] Mendées France M., Fractal geometry and analysis pp 325– (1989)
[12] Mendées France M., Bull. Soc. Math. Franco 109 (2) pp 207– (1981)
[13] Santalóo L. A., Integral geometry and geometric probability (1976)
[14] Sulanke R., Acta Math. Acad. Sci. Hungar. 17 pp 233– (1966) · Zbl 0144.44402
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.