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Source. Ball–Serra, published erratum (2011), printed p. 378, PDF p. 2. The following is a complete verification of its example and of the particular inference it refutes.

Choose a field containing $a\notin\{0,1\}$, for example $F=\mathbb Q$ and $a=2$. Set

$$S_1=\{0,1\},\quad S_2=\{0,a\},\quad D_1=\{1\},\quad D_2=\{a\},\quad f=X_1X_2(X_1-X_2).$$

The polynomial vanishes at every grid point with a zero coordinate, whereas

$$f(1,a)=a(1-a)\ne0.$$

Its nonzero grid values are therefore confined to $D_1\times D_2$, with a nonzero value there. The old exponent bounds would require a monomial with both exponents between one and one, namely $X_1X_2$. But

$$f=X_1^2X_2-X_1X_2^2$$

has no such monomial. This refutes the old corollary even after making its nonzero-grid-value hypothesis explicit.

To see the error in its proof, write

$$g_1=X_1(X_1-1),\qquad g_2=X_2(X_2-a).$$

The exact identity printed in the erratum is

$$f=g_1X_2-g_2X_1+(1-a)X_1X_2.$$

The last summand is the normal remainder modulo $(g_1,g_2)$, since its degree in each variable is less than two. Its coefficient of $X_1X_2$ is nonzero, but the two ideal terms cancel that monomial in $f$. Thus the coordinate upper bounds on the remainder do not transfer to an original monomial of $f$.

The erratum prints only $a\ne1$. To give the stated nonzero exceptional value and two distinct elements of $S_2$, one also needs $a\ne0$. This parameter qualification is made explicit here; it does not alter the counterexample or constitute a further author-issued erratum.

The valid conclusion keeps only the coordinate lower bounds. Both monomials displayed in $f$ satisfy those bounds. The corrected statement is recorded once on the canonical Corollary 4.2 page, and the coordinatewise reduction it needs is proved in the grid-ideal lemma.