Random walks on a line and algebraic curves

This work is devoted to the studying the generating function of the first hitting time of the positive semi-axis under the homogeneous discrete integer random walk on a line. In the first part of the work the increments are supposed to be independent. Recurrent relations for the probabilities allow us to write the system of equations for the sought generating function. Applying the resultants technique, we succeed to reduce this system to a single equation. Then we can study it by calculating the genus of the correposponding plane algrebraic curve via analyzing its singularities. In the work we write the sought equations for some random walks and we show that if the increments take all integer values from $-2$ to $2$, or from $-1$ to $3$ with equal probabilities or they take equally probable values $-1$ and $4$, then the curve is rational, while this is not true in the first case.
In the second part of the work we consider a symmetric process, the increments take the values $-1$, $0$, $1$, but then we suppose a non-zero correlation of each next increment with the previous one. For such process the equation for the generating function defines an elliptic curve depending on the square of the correlation coefficient for neighbouring increments if all increments are non-zero and it defines a hyperelliptic curve of genus $2$. The degeneration criterion of the latter is the presence of multiple roots of a sixth order polynomial under general symmetrically distributed conditional probabilities.