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I'm working on a problem that want me to solve for solutions given four equations that is equal to an integer,

For instance, consider the variables $a_1,a_2,\dots,a_m\in \mathbb{Z}_{\geq 0}$ $a_n\neq a_m$, $\max a_m \leq C$ and $M\in \mathbb{Z}_{\geq0}$ take $m=9,C=9$

and we have \begin{align} a_1+a_2+a_3 +a_{4} &=M\\ a_{2}+a_3+\dots +a_{6} &=M\\ a_4+a_5+\dots +a_{8} &=M\\ a_6+a_7+a_8+a_{9} &=M \\ \end{align} And I want to solve for each $a_m$, well for small $M$ I guess I can just compute by basic algebra and brute force. But I was thinking if there are some nice methods, I can use to solve for larger $M$, I tried looking into the partition function of integers with constrains but nothing I found that can relates to solve for values of these variables. If anyone can give me some ideas on how to approach or what topics I should read more to get an idea that does not involve brute force?

Thanks!

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    $\begingroup$ Is the first equation correct? The other three all have 5 terms, but that one only has 4. $\endgroup$
    – Eric Nathan Stucky
    Commented Jul 28, 2023 at 2:08
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    $\begingroup$ Certainly in general you are asking for the number of integer points in a lattice polytope, so Ehrhart theory may be helpful for you, for instance if you're interested in how these grow in $C$. Broadly speaking, combinatorial optimization may be helpful, but I don't know enough about that corner of the world to give suggestions. For your specific question, I don't see what the structure might be for other $m$ so I can't say too much. $\endgroup$
    – Eric Nathan Stucky
    Commented Jul 28, 2023 at 2:24
  • $\begingroup$ @EricNathanStucky Hi thanks for the comment, indeed I made a mistake, the first and fourth equation has 4 terms, middle two has 5 terms. $\endgroup$
    – Remu X
    Commented Jul 28, 2023 at 3:37

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