Properties of roots of recurrence relations in the context of exponential algorithms in order to decrease the upper bound of the running time
Let's start with your first question. You are considering two polynomials,
$$
P(x) = x^n - a_r x^{n-t_r} - a_{r-1} x^{n-t_{r-1}} - \cdots - a_1 x^{n-t_1}, \\
Q(x) = x^n - a_r x^{n-t_r} - \cdots - b_i x^{n-t_i} - \cdots - a_1 x^{n-t_1},
$$
where all $a_i$ are positive, and $0 &lt; b_i &lt; a_i$. Thus $Q$ results from replacing the term $-a_i x^{n-t_i}$ with $-b_i x^{n-t_i}$.

Let $\tau$ be the largest real root of $Q(x)$; note that $\tau &gt; 0$. Thus
$$ P(\tau) = Q(\tau) + (b_i - a_i) \tau^{n-t_i} &lt; 0. $$
On the other hand, $\lim_{x\to\infty} P(x) = \infty$. Therefore $P$ must have another root beyond $\tau$.

You can try attacking your second question using similar reasoning.