On fully split lacunary polynomials in finite fields
  We estimate the number of possible types degree patterns of $k$-lacunary polynomials of degree $t < p$ which split completely modulo $p$. The result is based on a combination of a bound on the number of zeros of lacunary polynomials with some graph theory arguments. 
Khodakhast Bibak and Igor E. Shparlinski
math.NT