I have the following exponential sums ($x\in\mathbb{R}$) $$f(x)=\sum_{i=1}^Na_iP_i(x)b_i^x$$ where $P(x)$ is some monomial, e.g., $x^2, x^3,\dots$, so $f(x)$ looks like $$f(x)=a_1x^3b_1^x+a_2x^2b_2^x+a_3x^3b_3^x+\dots+a_Nx^{k_N}b_N^x$$ I am interested in the possible number of roots ($f(x)=0$). I found a thesis by Joel D. Dreibelbis (Bounding Intersections of Orbit Sets with Curves) where he investigates such exponential sums. From that I know that if $b_i>0$ then the number of integer zeros is at most $\sum_{i=1}^N(\deg(P_i(x))+1)-1$.
However, I would like to have a better bound and for any zero (not just integer), possibly using some variant of Descartes rule of signs. My questions are:
- Is there some variant of the rule of signs for this case?
- How to order the monomials. According to the degree of $P_i(x)$ or the size of $b_i(x)$?