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2 Changed bounds in question to make it less trivial

I'm looking to efficiently zero-test "sparse integers", i.e. integers of the form $\sum C_i \cdot A_i^{X_i}$ (where $A_i, C_i, X_i$ are integers); equivalently test if a given (integer or rational) point is a zero of a sparse polynomial. For example, a randomised algorithm would be to compute the sum modulo a random prime since Fermat's Little Theorem reduces computations to a "manageable" level and with good probability there won't be a false positive. I'm wondering if this can be derandomised.

So beside the obvious does anyone know anything relevant about the general problem, I have more of a first step question: Is (are) there (infinitely many) positive integer(s) $M$ such that there are integers $A,B,C,D,E,F,X,Y,Z$ all with absolute value $> |A|,|B|,|C| < M$, $D,E,F,X,Y,Z>M$ and $$0 < A\cdot D^X+B\cdot E^Y+C \cdot F^Z< \log M?$$

1

abc-conjecture meets Catalan conjecture?

I'm looking to efficiently zero-test "sparse integers", i.e. integers of the form $\sum C_i \cdot A_i^{X_i}$ (where $A_i, C_i, X_i$ are integers); equivalently test if a given (integer or rational) point is a zero of a sparse polynomial. For example, a randomised algorithm would be to compute the sum modulo a random prime since Fermat's Little Theorem reduces computations to a "manageable" level and with good probability there won't be a false positive. I'm wondering if this can be derandomised.

So beside the obvious does anyone know anything relevant about the general problem, I have more of a first step question: Is (are) there (infinitely many) positive integer(s) $M$ such that there are integers $A,B,C,D,E,F,X,Y,Z$ all with absolute value $> M$ and $$0 < A\cdot D^X+B\cdot E^Y+C \cdot F^Z< \log M?$$