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On the number of real roots of a random algebraic equation. II

Published online by Cambridge University Press:  24 October 2008

J. E. Littlewood  and
A. C. Offord
J. E. Littlewood
Affiliation:
Trinity CollegeCambridge
A. C. Offord
Affiliation:
St John's CollegeCambridge
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Extract

An equation with real coefficients and given degree n being selected at random, about how many real roots may it be expected to have? The present series of papers is concerned with this question and matters arising out of it. The results we have arrived at were stated without proof in our paper I (with the same general title), which contains also some introductory remarks to which we may refer the interested reader. Here we summarize as follows.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 35 , Issue 2 , April 1939 , pp. 133 - 148
Copyright
Copyright © Cambridge Philosophical Society 1939

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References

† All equations occurring in the present paper have real coefficients, and “equation” is to be understood throughout in this sense.

‡ J. London Math. Soc. 13 (1938), 288.Google Scholar

† The difference between types E and H may be ignored for our present purposes.

‡ Bloch, A. and Pólya, G., Proc. London Math. Soc. (2), 33 (1932), 102–14.CrossRefGoogle Scholar

§ Schmidt, E., S.B. preuss. Akad. Wiss. (1932), 321.Google ScholarSchur, I., S.B. preuss. Akad. Wiss. (1933), 403–28.Google ScholarSzegö, G., S.B. preuss. Akad. Wiss. (1934), 86–98.Google Scholar These authors, as also Bloch and Pólya, prove general theorems of which the result quoted is a very special case.

∥ Who proved that the maximum number was at any rate o(n), and that the average number was O(√n).

The result O(√n) for type E is due to Schmidt. The theorem of Schur yields slightly less, namely O(√(n log n)), but is more precise in other ways. For a simple proof of this O(√(n log n)) result see the note at the end of the present paper.

** By “most equations” we mean all but a proportion o(1) of the whole aggregate.

† Or, for that matter, a proportion n −10, or any particular negative power.

† Our arguments bear sometimes on complex and sometimes on real zeros; for clarity we use the term “(real) roots” in the latter case.

† We have p ≥ 1 on account of n ≥ 16.

† The exceptional sets are rather smaller than in case E.

† Cf., for example, Cramér, , “Random variables and probability distributions”, Cambridge Tracts, no. 36 (1937), p. 23.Google Scholar

† Cramér, op. cit. p. 36.

‡ Paley, Due to and Zygmund, , Proc. Cambridge Phil. Soc. 28 (1932), 190–205 (192).CrossRefGoogle Scholar Our original proof was different, and we are indebted to Prof. Zygmund for drawing our attention to the advantages of using the lemma.

§ Khintchine, , Math. Z. 18 (1923), 109–16 (111).CrossRefGoogle Scholar

† The choice of k secures that the range X m is part of the total summation of ν for the relevant range ½12k≥k≥k of m.