On real roots of systems of trancendental equations with real coefficients

The work is devoted to the study of the number of real roots of systems of transcendental equations in $\mathbb C^n$ with real coefficients, consisting of entire functions, in some bounded multidimensional domain $D\subset \mathbb R^n$. It is assumed that the number of roots of the system is discrete (then it is no more than countable). For some entire function $\varphi (z), z\in \mathbb C^n$, with real Taylor coefficients at $z=0$, and a given system of equations, the concept of a resultant $R_\varphi(t)$ is introduced, which is an entire function of one complex variable $t$. It is constructed using power sums of the roots of the system in a negative degree, found using residue integrals. If the resultant has no multiple zeros, then it is shown that the number of real roots of the system in $D=\{x\in \mathbb R^n: a<\varphi(x)<b\}$ ($x=\mathrm{Re}\, z $) is equal to the number of real zeros of this resultant in the interval $(a,b)$. An example is given for a system of equations.