Let $f_j(x)$, $j=1,\dots,N$ be functions which are sufficiently regular. (This can be adjusted accordingly but let's assume that these functions are nice.) Furthermore, suppose that for each of these functions, one can find $x$ such that $f_j(x) = 0$. However, the roots of $f_j(x)$ and $f_k(x)$, $j \neq k$ need not be equal. Furthermore, some $f_k$'s have more roots than others.
Is anyone familiar with literature on which the roots of linear combinations of $Z_1 f_1(x) + \dots + Z_N f_N(x) = 0$ have been studied? I am mostly curious about how the number of distinct roots of the above linear combination is related to the coefficients $Z_1,\dots,Z_N$.
Suggestions are appreciated!
Let me assume your nice functions are real polynomials of degree $\le N$. One can easily choose to have roots of particular kind using polynomials of exact degree N and yet form a basis for (the vector space of all real) polynomials of degree $\le N$. (so constants and polynomials of every possible degree upto N are an the linear combinations)
And so among them linear combinations you will be able to find polynomials with no real root, with all real roots and everything in the spectrum in between.
Among non-polynomials $\cos^2 x$ and $\sin^2 x$ have the constant 1 among its linear combinations, whereas $\cos^2x - \sin^2 x$ has many roots being $\cos 2x$