Note that these are simply a special case (generalization? depending on terminology) of exponential polynomials. Or rather, to be more accurate, they can be treated as such. Let $t=\ln x$, and then your monomials become $a_i e^{ty_i}$. The change of variables does introduce some technical issues, obviously (in dimension $n$, you need to do it separately for each of the $2^n$ quadrants, and then possibly iterate on the coordinate zero hyperplanes).

Over the complexes, as was pointed out by Gerry, there is potentially infinitely many roots. Over the reals, the number of roots is still finite, and some of the elementary results like Descartes's rule of signs do translate fairly well to that setting. (The fact that for ordinary polynomials, the derivative is either 0 or has fewer roots is not so crucial in many applications).

In several variables, the theory of fewnomials developed by Khovanskii gives some (pessimistic) upper bounds on the number of real roots (it also gives some restrictions on the patterns of the complex roots).

None of this is overly difficult, but I don't see the point of going into more details without a more specific question.

2 Clarified (?) the change of variable issue

Note that these are simply a special case of exponential polynomials. Or rather, to be more accurate, they can be treated as such. Let $t=\ln x$, and then your monomials become $a_i e^{ty_i}$. The change of variables does introduce some technical issues, obviously (in dimension $n$, you need to do it separately for each of the $2^n$ quadrants, and then possibly iterate on the coordinate zero hyperplanes).

Over the complexes, as was pointed out by Gerry, there is potentially infinitely many roots. Over the reals, the number of roots is still finite, and some of the elementary results like Descartes's rule of signs do translate fairly well to that setting. (The fact that for ordinary polynomials, the derivative is either 0 or has fewer roots is not so crucial in many applications).

In several variables, the theory of fewnomials developed by Khovanskii gives some (pessimistic) upper bounds on the number of real roots (it also gives some restrictions on the patterns of the complex roots).

None of this is overly difficult, but I don't see the point of going into more details without a more specific question.

1

Note that these are simply a special case of exponential polynomials. Over the complexes, as was pointed out by Gerry, there is potentially infinitely many roots. Over the reals, the number of roots is still finite, and some of the elementary results like Descartes's rule of signs do translate fairly well to that setting. In several variables, the theory of fewnomials developed by Khovanskii gives some (pessimistic) upper bounds on the number of real roots (it also gives some restrictions on the patterns of the complex roots).

None of this is overly difficult, but I don't see the point of going into more details without a more specific question.