I was able to reduce a research problem relating to normal numbers to analyzing the solutions of the following family of nonlinear systems of equations: \begin{align*} c_0+c_1+\cdots+c_{t-1}&=2t\\ c_0c_1+c_1c_2+\cdots+c_{t-2}c_{t-1}&=2t\\ c_0c_1c_2+c_1c_2c_3+\cdots+c_{t-3}c_{t-2}c_{t-1}&=2t\\ \cdots\\ c_0\cdots c_{t-2}+c_1\cdots c_{t-1}&=2t\\ c_0\cdots c_{t-1}&=2t, \end{align*} where $t \geq 3$ is an integer. I should remark that the expressions on the left hand side are NOT symmetric sums. For example, the second line does not have the term $c_0c_2$. As a result there does not seem to be very much symmetry. I need to prove that there exists a solution $(c_0,c_1,\cdots,c_{t-1})$ where $c_0 \in [t,t+1]$ and $c_i \in \left[1+\frac {1} {2t},1+\frac {1} {t-1}\right]$ for $i \in [1,t-1]$. I have verified this by computer up to around $t=100$ and I have some hand wavy reasons to suspect that this system will always have such a solution. I'm really at a loss of what to do and would appreciate any suggestions.

Now I can prove an even stronger and more interesting theorem if a more general nonlinear system can be analyzed. Consider the system of equations \begin{align*} c_0+c_1+\cdots+c_{t-1}&=2t+\epsilon_1\\ c_0c_1+c_1c_2+\cdots+c_{t-2}c_{t-1}&=2t+\epsilon_2\\ c_0c_1c_2+c_1c_2c_3+\cdots+c_{t-3}c_{t-2}c_{t-1}&=2t+\epsilon_3\\ \cdots\\ c_0\cdots c_{t-2}+c_1\cdots c_{t-1}&=2t+\epsilon_{t-1}\\ c_0\cdots c_{t-1}&=2t+\epsilon_t, \end{align*} where $t\geq 3$ is an integer and the terms $\epsilon_i$ are "small". I want to prove that this modified system has a solution "close" to the desired solution of my original system (I can't be much more precise at this point and there is a lot of flexibility in terms of what I can work with). If I can get very precise estimates of how much the solutions $(c_0,\cdots,c_{t-1})$ vary depending on $(\epsilon_0,\cdots,\epsilon_{t-1})$, then I might be able to prove an even stronger theorem.

I should remark that there are many seemingly promising identities that can be derived from the original system that unfortunately don't seem to help. For example, it is not difficult to show that $\frac {1} {c_0}+\frac {1} {c_{t-1}}=1$. Also there is some flexibility in my original problem and there are other possible systems that might be useful to analyze, but so far I haven't come across any families of systems that look more promising than the one I'm asking about here. I've also been unable to use any theorems I know from analysis to get anywhere on this problem. I'm not sure if my tags are appropriate as I don't know really what would go into solving this problem.


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