# zbMATH — the first resource for mathematics

Algebraic polynomials with non-identical random coefficients. (English) Zbl 1048.60042
Summary: There are many known asymptotic estimates for the expected number of real zeros of a random algebraic polynomial $$a_0 +a_1 x+ a_2 x^2+ \cdots +a_{n-1}x^{n-1}.$$ The coefficients $$a_j$$ $$(j=0, 1, 2, \dots, n-1)$$ are mostly assumed to be independent identical normal random variables with mean zero and variance unity. In this case, for all $$n$$ sufficiently large, the above expected value is shown to be $$O(\log n)$$. Also, it is known that if the $$a_j$$ have non-identical variance $$\binom{n-1}{j}$$, then the expected number of real zeros increases to $$O(\sqrt{n})$$. It is, therefore, natural to assume that for other classes of distributions of the coefficients in which the variance of the coefficients is picked at the middle term, we would also expect a greater number of zeros than $$O(\log n)$$. For two different choices of variance for the coefficients we show that this is not the case. Although we show asymptotically that there is some increase in the number of real zeros, they still remain $$O(\log n)$$. In fact, so far the case of $$\text{var}(a_j)={\binom{n-1}{j}}$$ is the only case that can significantly increase the expected number of real zeros.

##### MSC:
 60G99 Stochastic processes 60H99 Stochastic analysis
Full Text:
##### References:
  A. T. Bharucha-Reid and M. Sambandham, Random polynomials, Probability and Mathematical Statistics, Academic Press, Inc., Orlando, FL, 1986. · Zbl 0615.60058  Alan Edelman and Eric Kostlan, How many zeros of a random polynomial are real?, Bull. Amer. Math. Soc. (N.S.) 32 (1995), no. 1, 1 – 37. · Zbl 0820.34038  Kambiz Farahmand, Topics in random polynomials, Pitman Research Notes in Mathematics Series, vol. 393, Longman, Harlow, 1998. · Zbl 0949.60010  K. Farahmand, On random algebraic polynomials, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3339 – 3344. · Zbl 0933.60081  K. Farahmand and P. Hannigan, The expected number of local maxima of a random algebraic polynomial, J. Theoret. Probab. 10 (1997), no. 4, 991 – 1002. · Zbl 0897.60055 · doi:10.1023/A:1022618801587 · doi.org  K. Farahmand and P. Hannigan, Large level crossings of random polynomials, Stochastic Anal. Appl. 20 (2002), no. 2, 299 – 309. · Zbl 0997.60059 · doi:10.1081/SAP-120003436 · doi.org  K. Farahmand and M. Sambandham. Real zeros of classes of random algebraic polynomials. J. Appl. Math. and Stochastic Analysis, 16:249-255, 2003. · Zbl 1051.60057  M. Kac. On the average number of real roots of a random algebraic equation. Bull. Amer. Math. Soc., 49:314-320, 1943. · Zbl 0060.28602  B. F. Logan and L. A. Shepp, Real zeros of random polynomials, Proc. London Math. Soc. (3) 18 (1968), 29 – 35. · Zbl 0245.60047 · doi:10.1112/plms/s3-18.1.29 · doi.org  B. F. Logan and L. A. Shepp, Real zeros of random polynomials. II, Proc. London Math. Soc. (3) 18 (1968), 308 – 314. · Zbl 0177.45201 · doi:10.1112/plms/s3-18.2.308 · doi.org  S.O. Rice. Mathematical theory of random noise. Bell System Tech. J., 25:46-156, 1945. Reprinted in: Selected Papers on Noise and Stochastic Processes (ed. N. Wax), Dover, New York, 1954, 133-294. · Zbl 0063.06487  M. Sambandham, On a random algebraic equation, J. Indian Math. Soc. (N.S.) 41 (1977), no. 1-2, 83 – 97. · Zbl 0437.60049  M. Sambandham, On the average number of real zeros of a class of random algebraic curves, Pacific J. Math. 81 (1979), no. 1, 207 – 215. · Zbl 0411.60066  J. Ernest Wilkins Jr., An asymptotic expansion for the expected number of real zeros of a random polynomial, Proc. Amer. Math. Soc. 103 (1988), no. 4, 1249 – 1258. · Zbl 0656.60062
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.