Let $K$ be a field, and let $G = (g_1, \ldots, g_m)$ be a Gröbner basis in $K[x_1, \ldots, x_n]$ (i.e. $G$ is a Gröbner basis for the ideal it generates).
By Adams, Loustaunau - An Introduction to Gröbner Bases, Theorem 1.6.7, for any polynomial $f$, the remainder of the division (reduction) of $f$ with $G$ is unique: i.e. if $f = \sum q_ig_i + r$, where $q_i$ are the quotients obtained from the division algorithm, then $r$ is unique.
Adams and Loustaunau note that the quotients $q_i$ are not unique.
My question is: under what hypothesis do we have uniqueness of the quotients?
In particular: if the Gröbner basis consists of two elements: $G = (g_1, g_2)$, when are the quotients $q_1, q_2$ unique?
An example: if $g_1(x,y) = y$ and $g_2(x,y) = g(x)$ is a polynomial depending only on $x$, then I believe that $q_1, q_2$ are unique (because $K[x]$ is a UFD).