SELECTED CHAPTERS FROM ALGEBRA I. R. Shafarevich Preface

28 I. R. Shafarevichthey do not increase. If they are again represented by the points (i c i ) then theywill have \one hump" (Fig. 8).Fig. 8For example, polynomial (1 + x) n is reciprocal: this follows from the propertyCnm = Cn;m n of binomial coecients (see Section 3 of Chapter II). It is also unimodal:this follows from the property of binomial coecients proved in Section 3of Chapter II.It can be proved that if the polynomials f(x) andg(x) have nonnegative coecients,if they are reciprocal and unimodal, then f(x)g(x) is unimodal. Theproof is quite elementary, but a little involved. From this theorem it follows thatthe polynomial G(x) is unimodal. However, you can easily prove yourself this specialcase (Problem 3). Now, it is easy to determine the term with the greatestcoecient in a reciprocal unimodal polynomial. Namely, if the term c k x k has thegreatest coecient, since the polynomial is reciprocal we have c n;k = c k and thereis the symmetric term c k x n;k . We can take that k 6 n=2 and n ; k > n=2.Since the polynomial is unimodal, none of the terms c i x i where k 6 i 6 n ; k canhave smaller coecient, for otherwise there would be two \humps" on the graph.Hence, the greatest coecient must be the middle coecient c n=2 if n is even ortwo \equally middle" coecients c n;1 = c n+1 if n is odd (though there may be22other coecients equal to them). In particular, we see that if n is even, then inG(x) the term xterms of G(x), x5n2 has the greatest coecient, and if n is odd then there are two5n;125n+12and xwith equal greatest coecients.In the polynomial F (x) this term is multiplied by x n and has degree 5n7n2 + n =if n is even. If n is odd, there are two terms with equal coecients with degree25n ; 1+ n = 7n ; 1 5n +1 7n +1and + n = . Therefore, if dice is thrown n22 22times the most probable score is 7n if n is even, and if n is odd there are two scores2which are both most probable: 7n ; 1 7n +1and .2 2Consider one more problem of the same type. A certain quantity ofm physicalparticles are registered by n instruments, so that each particle can be registeredby any instrument, and the registration of a particle by all instruments are taken