# The behavior of the roots of some polyonimial appearing in the study of the dying rabbit problem

Given natural numbers $h,k\geq1$ we define the following polynomial: $$g_{k,h}(x)=x^{k+h-1}-x^{k-1}-\dots-x-1.$$ These polynomials are important in the study of dying rabbit problems, since they determine the expression of the solution.

Denote $f_{k,h}(x) = (x−1)g_{k,h}(x) = x^{k+h}−x^{k+h−1}−x^{k+1}$.

Antonio Antonio M. Oller [1] studied , in some sense, the behavior of the roots of this polynomial in terms of $k$ and $h$. Let $Z(f)$ denote the set of complex roots of $f$.

Proposition 5. All the roots of $g_{k,h}$ are distinct.

Proposition 7. For every $w \in Z(g_{k,h})$, the equalities $|w| = \alpha_{k,h}$ and $w = \alpha_{k,h}$ are equivalent.

The proof of Proposition 5 is based on Lemma 2.(c).

Lemma 2.(c). The polynomial $g_{k,h}(x)$ has no complex root with modulus in the interval $(1, \alpha_{k,h})$.

Note that Lemma 1 state that if $k>1$, $g_{k,h}(x)$ has a unique positive real root $\alpha_{k,h}$ which lies in $(1,2)$ This can be done easily by applying Descartes’ rule of signs and observe that $g_{k,h}(1) < 0 < g_k,h(2)$.

Question:

By computer calculation I find that the Lemma 2 c) is incorrect, since these polynomials indeed have imaginary roots with modulus greater than one by numerical calculations of the roots. For example that the roots of $g_{2, 3}=-1 - x + x^4$ are $${-0.724492, -0.248126 - 1.03398 I, > -0.248126 + 1.03398 I, 1.22074},$$ where the modulus of the pair of the second root and the third is 1.06334.

Another thing I want to mention is that the statement the only real roots of $g_{k,h}$ are $\alpha_{k,h}$ and $-1$ (only if $k$ is odd and $h$ is even)'' in the fourth line the proof of Proposition 7 is wrong, for if $k$ is even and $h$ is odd, then the coefficients of $f_{k,h}(-x)$ have only one sign change, and $g_{k,h}(x)$ has one negative zero by Descartes' rule of signs. In fact, this zero is larger then $-1$, easily by observing that $f_{k,h}(-1)<0$ and $f_{k,h}(0)>0$ .

References

[1]Antonio M. Oller-Marcén, The Dying Rabbit Problem Revisited, INTEGERS, 9 (2009), 129-138.

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What is your question? Are you simply announcing that you found a counterexample to a lemma? –  Douglas Zare Mar 28 at 8:26