Nonexistence of degree bounds of various bases for ideals of polynomials over the integers

It has been proved in [L] that there is no universal degree bound of Gröbner bases when the coefficient ring is Z, where Gröbner bases are defined as in [T]. In this paper we prove that the same result is true for each of the following constructive bases: (1) Gröbner bases in the sense of [Bu]; (2) Gröbner bases in the sense of [KK]; (3) detaching bases in the sense of [A]; (4) Szekeres basis in the sense of [S]. We also discuss several open problems about degree bounds, which are motivated by our examples as well as by results from commutative algebra.