# Lower bound on diophantine system of inequalities with all but one non-linear constraint

I have a system of $n+1$ diophantine inequalities, in the following form:

$$f_{1}(x_1, x_2, \dots, x_n) \geq 0$$

$$f_{2}(x_1, x_2, \dots, x_n) \geq 0$$

$$\vdots$$

$$f_{m}(x_1, x_2, \dots, x_n) \geq 0$$

where $f_{1}(x_1, x_2, \dots, x_n)$ is a $n$-variate polynomial in $(x_{1},\dots, x_{n})\in\mathbb{Z}_{n}$. Whereas, $f_{2}(x_1, x_2, \dots, x_n)$, $\dots,$ $f_{n}(x_1, x_2, \dots, x_n)$ are all linear in $(x_1, x_2, \dots, x_n)$ . Moreover, there are more constraints than variables, i.e, $m > n$ (in fact, it is roughly like $m \sim 2n$, if that helps).

Can anyone please reference me to places where I could find answers to:

1. Given such a (above) set of inequalities, can I determine if the system is consistent ? (I have seen consistency checks mentioned for linear systems, but here one of the constraints is non-linear.)

2. If there any sort of rough lowerbound for number of (integer) solutions for $(x_1, x_2, \dots, x_n)$ ?

3. If there is no such general to state lowerbound (to the number of (integer) solutions for $(x_1, x_2, \dots, x_n)$), is there a systematic method to compute such a lowerbound ?

4. If there is a lowerbound / upperbound for the number of (integer) solutions to $(x_1, x_2, \dots, x_n)$, to the linear part of the above system, i.e, for:

$$f_{2}(x_1, x_2, \dots, x_n) \geq 0$$

$$\vdots$$

$$f_{n}(x_1, x_2, \dots, x_n) \geq 0$$

can I comment about the lowerbound for number of solutions to the entire system (along wit the non-linear function $f_{1}(x_1, x_2, \dots, x_n)$) ?

Thanks.

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