Constant problem for discrete functions Is there an algorithm to decide whether a closed-form expression over integer variables using, say, $\{+,-,\times,\div,\text{^},\lfloor\text{lg}\rfloor,!,()\}$, or some other useful set of operators, is everywhere zero?

For example, is $(jk)^2 +k!-2j$ always 0 when $j,k$ are integers (obviously not)? If some of the operators make it impossible, is it possible with just a subset, for example if one removes factorial?
To decide if a particular diophantine equation (i.e., an expression in integers using multiplication and addition/subtraction equated to zero) has any solutions is undecidable. This is essentially the answer to the negative to Hilbert's tenth problem, which asks for an algorithm to determine if a diophantine equation has a solution.