# Existence of solutions to a Specific kind of non-linear system of equations with rational variables

Consider a system of the form

$c_{11}a_1b_{1} + c_{12}a_2b_{2} + \cdots c_{1n}a_nb_{n} = q_1$

$c_{21}a_1b_{1} + c_{22}a_2b_{2} + \cdots c_{2n}a_nb_{n} = q_2$

$\vdots$

$c_{m1}a_1b_{1} + c_{m2}a_2b_{2} + \cdots c_{mn}a_nb_{n} = q_m$

$c_{11}a_1b_{1} + c_{12}a_2b_{2} + \cdots c_{1n}a_nb_{n} \neq q_1$

$c_{21}a_1b_{1} + c_{22}a_2b_{2} + \cdots c_{2n}a_nb_{n} \neq q_2$

$\vdots$

$c_{m1}a_1b_{1} + c_{m2}a_2b_{2} + \cdots c_{mn}a_nb_{n} \neq q_m$,

where the $c_{ij}$ are 0 or 1 (denoting presence/absence of a term), the $a_j$ and $b_{j}$ are rational number variables in $[0,1]$ (the unknowns) and the $q_i$ are constant rationals in $[0,1]$.

I would like to know whether there is a standard algorithm for solving such systems of equations? And i would like to know whether such systems (as described above) are decidable, that is, whether there is a procedure for deciding whether a solution exists for such a system?

What about when the system excludes disequations (no $\neq$): Is there a method -- besides Newton's method with Jacobians for general nonlinear systems -- which is decidable?

Could someone point me to literature which might answer my questions?

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