Euclidean distance degree of complete intersections via Newton polytopes

In this note, we consider a complete intersection $X=\{x\in {\mathbb{R}}^n:f_1(x)=\dots =f_m(x)=0\},n>m$ and study its Euclidean distance degree in terms of the mixed volume of the Newton polytopes. We show that if the Newton polytopes of $f_j,j=1,\dots ,m$ contain the origin then when these polynomials are generic with respect to their Newton polytopes, the Euclidean distance degree of $X$ can be computed in terms of the mixed volume of Newton polytopes associated to $f_j$. This is a generalization for the result by P. Breiding, F. Sottile and J. Woodcock in case $m=1$.