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\def\cx{{\bf C}}
{\bf Ideals, Varieties and Algorithms}, {\it first edition}
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{\sl Errata for the third printing as of July 14, 2000}
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Page 29, line $4$: Replace ``were'' with ``where''
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Page 54, line 17: Replace ``term orderings'' with ``monomial
orderings''
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Page 58, line $-1$: Replace ``multideg'' with ``multidegree''
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Page 70, line 13: Replace ``$x^\alpha$'' with ``$x_1^\alpha$''
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Page 73, part b of Exercise 12: ``the gradlex order $>_{gradlex}$''
should be ``the grlex order $>_{grlex}$''
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Page 82, line $-2$: Replace ``proposition'' with ``lemma''
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Page 99, line 9: Replace displayed equation with
``$-(4/3)x^3z+x^2y^2+2xyz-(4/3)y^3-(1/3)z^2 = 0$''
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Page 110, line 15: Replace ``Exericse'' with ``Exercise''
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Page 111, line 15: Replace ``in in'' with ``in''
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Page 112, line $-7$: Replace ``167383594'' with ``170255391''
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Page 127, line $-5$: Replace ``thus'' with ``Thus''.
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Page 144, line 1: ``\S7 of Chapter 2'' should be ``\S8 of Chapter 2''
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Page 144, line $-16$: Replace ``need to show that notion'' with ``We
need to show that the notion''
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Page 148, line 9: Replace ``$f$ and $g$'' with ``$g$ and $h$''
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Page 180, line 1: ``$xz$ and dividing $xz$ by $xy$ gives 0 plus
remainder $xy$'' should be ``$xy$ and dividing $xz$ by $xy$ gives 0
plus remainder $xz$''
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Page 183, line 5: ``If $I$ and $J$ be ideals'' should be ``If $I$ and
$J$ are ideals''
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Page 196, line 3: Replace ``repect'' with ``respect''
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Page 207, line $-10$: Replace ``irreducible varieties''
with ``varieties''
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Page 232, line 12: Replace ``$k[x_1,\dots,x_n]/I$'' with
``$\cx[x_1,\dots,x_n]/I$''
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Page 232, line $-7$: Replace ``$k[x_1,\dots,x_n]/I$'' with
``$\cx[x_1,\dots,x_n]/I$'' twice
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Page 232, line $-6$: Replace ``$k[x_1,\dots,x_n]/I$'' with
``$\cx[x_1,\dots,x_n]/I$''
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Page 234, line $-18$: Add the following: ``Hint: Use Proposition 4
and part (iii) of Theorem 6.''
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Page 244, line 20: Replace ``{\it smooth\/}'' with
``{\it nonsingular\/}''
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Page 246, line $-1$: Replace ``Exercise 4'' with ``Exercise 2''
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Page 274, line 4: Replace ``Exercise 13'' with ``Exercise
14''
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Page 279, lines 5, 9 and 10: Replace ``$x_m$'' with
``$x_n$''
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Page 291, line $-4$: Replace ``${u_2^2\over u_1^2+u_2^2}$'' with
``${u_1u_2^2\over u_1^2+u_2^2}$''
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Page 299, line $-3$: Replace ``${\scriptstyle{\rm LC}}(h,x_n)$'' with
``${\scriptstyle{\rm LC}}(f_n',x_n)$''
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Page 315, line 2: replace
``$\sum_{i=0}^k h_k(x_k,\ldots,x_n) \sigma_i(x_1,\ldots,x_n)$'' with
the corrected formula
``$\sum_{i=0}^k (-1)^i h_{k-i}(x_k,\ldots,x_n)
\sigma_i(x_1,\ldots,x_n)$''
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Page 317, line $-8$: Replace ``$M_{\tau\nu}$'' with
``$M_{\nu\tau}$''
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Page 317, line $-7$: Replace ``$\tau\nu$ is the
permutation that takes $i$ to $\tau(\nu(i))$'' with ``$\nu\tau$ is the
permutation that takes $i$ to $\nu(\tau(i))$''
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Page 347, line $-6$: Replace ``it coordinates'' by ``its coordinates''
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Page 376, line $-6$: Replace ``grlex'' with ``grevlex''
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Page 393, lines 1 and 2 of Exercise 7: Replace ``$F_1 =
u+vy$ and $F_2 = u^2+u^2y$'' with ``$F_1 = u-vy$ and $F_2 =
u^2-v^2y$''
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Page 393, line 2 of part a of Exercise 7: Replace
``$y(1+y)$'' with ``$y(1-y)$''
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Page 393, line 1 of part b of Exercise 7: Replace
``$u^2y(1+y)$ and $v^2y(1+y)$'' with ``$u^2y(1-y)$ and $v^2y(1-y)$''
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Page 393, line 2 of part b of Exercise 7: Replace
``$uy(1+y)$, $vy(1+y)$ and $vy^2(1+y)$'' with ``$uy(1-y)$ and
$vy(1-y)$''
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Page 393, line 1 of part c of Exercise 7: Replace
``$\langle y^2(1+y)\rangle$'' with ``$\langle y(1-y)\rangle$''
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Page 421, line $-12$: Replace ``$5s-3$'' with ``$5s-5$''
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Page 421, line $-10$: Replace ``$(5s-3) + 4 = 5s + 1$''
with ``$(5s-5) + 4 = 5s - 1$''
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Page 425, lines 4 and 5: Replace with the following:
``where $\beta = \sum_{i \notin \{j_i,\ldots,j_r\}} b_i
e_i$. If $A \ne B$ and $A\cap B \ne \emptyset$, then show that
$$
[e_{i_1},\ldots,e_{i_m}] \ne [e_{j_1},\ldots,e_{j_r}]
$$
and that $A \cap B$ is a translate of''
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Page 425, line 7: Replace ``$\le \min(m,r)$'' with ``$<
\max(m,r)$''
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Page 458, line $-2$: Replace ``$z$-axis'' with
``$y$-axis''
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Page 459, line $-15$: Replace ``$x$-axis'' with
``$y$-axis''
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\end