# Interesting zeros of polynomials

Many sequences pn(x) of polynomials seem to have interesting sets of (complex) zeros. Here are some examples.

1. Let Kn(x) denote the sum over all connected simple graphs on an n-element vertex set of xe(G)-n+1, where e(G) denotes the number of edges of G. For instance, K3(x) = 3 + x. Zeros of K14(x): ps or pdf.

2. Let Xm,n(q) denote the chromatic polynomial of the complete bipartite graph Km,n. Equivalently, Xm,n(q) is the coefficient of xmyn/m!n! in the power series expansion about x=y=0 of (ex + ey -1)q. Zeros of X20,20(q): ps or pdf. Note. Appearances to the contrary, the integers 2, 3, 4, 5, 6, 7, 8, 9 are not zeros of X20,20(q). For instance, the zero close to q=2 is actually 2.0000000000000000000000000000000000000391710202489...

3. Let Cn(q) denote the sum of qA(P)over all lattice paths P from (0,0) to (n,n) with steps (0,1) and (1,0), such that P never rises above the line y=x, and where A(P) is the area under the path (and above the x-axis). The number of such paths is the Catalan number Cn, and Cn(q) is a standard q-analogue of Cn (see Enumerative Combinatorics, vol. 2, Exercise 6.34(a)). For instance, C3(q) = 1 + q + 2q2 + q3. Zeros of C32(q): ps or pdf.

4. Let Hn(r) denote the number of n×n matrices of nonnegative integers such that every row and column sums to r. For instance, Hn(0)=1, Hn(1)=n!, and H2(r)=r+1. It is known that for fixed n, Hn(r) is a polynomial in r of degree (n-1)2. (See for instance Enumerative Combinatorics, vol. 1, Proposition 4.6.19.) Zeros of H9(r): ps or pdf.

5. (suggested by G. Labelle) Let Nn(x) be the sum of (1/n)φ(d)xn/d over all (positive) divisors d of n, where φ(d) denote Euler's phi-function. If x is a positive integer, then Nn(x) is the number of "necklaces" of length n with x colors, i.e., the number of sequences, up to cyclic shift, with n terms from an alphabet of x letters. Zeros of N120(x): ps or pdf. Similar behavior is seen when φ is replaced by μ (the Möbius function of number theory), in which case necklaces are replaced by primitive necklaces (those with no symmetry).

6. (suggested by H. Wilf) Let Bn(x) denote the nth Bernoulli polynomial. Zeros of B300(x): pdf (corrected on 5/2/04). Click here (pdf file) for information on the real zeros of Bn(x). For the limiting curve formed by the complex zeros, click here.

7. Let pk(n) denote the number of partitions of the positive integer n into k parts. Let Fk(x) denote the sum over all n of pk(n)xn. For instance, F4(x) = x4 + x3 +2x2+ x. Zeros of F200(x): ps or pdf.

8. Zeros of the sum from 1 to 300 of xn/(sin n).