The complexity of solving a system of equations of the same degree

Many systems of interest in cryptography consist of equations of the same degree. Under the assumption that the degree of regularity is finite, we prove upper bounds on the degree of regularity of a system of equations of the same degree, with or without adding the field equations to the system. The bounds translate into upper bounds on the solving degree of the systems, and hence on the complexity of solving them via Gröbner bases methods. Our bounds depend on the number of equations in the system, the number of variables, and the degree of the equations.