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{\bf Ideals, Varieties and Algorithms}
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{\sl Errata for the second printing as of April 3, 1998}
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Page 5, line $-2$: ``$= \dots =$'' should be ``$= \cdots =$''
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Page 20, line 8: ``at this point is given by'' should be
``at the point given by''
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Page 24, part a of Exercise 8: ``0, 1, 2.~or 3'' should be
``0, 1, 2, or 3''
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Page 33, line $-1$: ``$f_n$'' should be ``$f_s$'' (twice)
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Page 40, line $-9$: ``the generator of an ideal'' should be ``the
generator of an ideal in $k[x]$''
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Page 53, line 9: ``$x^2 \cdot$ (leading term of $g$)''
should be ``$x^3 \cdot$ (leading term of $g$)''
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Page 54, line $-3$: ``is $S$'' should be ``in $S$''
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Page 65, line $-9$: ``$f_i$'' should be ``$f_s$''
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Page 70, statment of Theorem 5: ``$x_\alpha$'' should be ``$x_n$''
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Page 71, proof of Corollary 6: ``$n\alpha_0$'' and
``$(n+1)\alpha_0$'' should be ``$m\alpha_0$'' and ``$(m+1)\alpha_0$''
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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 79, Exercise 5: ``initial form'' should be ``leading
term''
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Page 80, Exercise 13: ``$N\ \ge\!1$'' should be ``$N \ge 1$''
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Page 81, part (i) of Proposition 1: ``by one of'' should be
``by any of''
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Page 81: The sentence ``In particular \dots algorithm''
which follows the statement of Proposition 1 should be part of the
statement of the proposition.
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Page 81, second paragraph of the proof: ``by one of'' should
be ``by any of''
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Page 83, line $-5$: ``leading term 1'' should be ``leading coefficient
1''
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Page 84, line $-8$, ``$\max(m(1),\dots,m(s))$'' should be
``$\max(m(1),\dots,m(t))$''
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Page 89, line $-11$: ``adjoined to $G'$'' should be
``adjoined to $G$''
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Page 94, line 4 of Example 1: ``$x^2y^2-z^3$'' should be
``$-x^2y^2+z^3$''
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Page 106, paragraph beginning ``From equation (3)'': The
``(3)'' should be ``(4)''
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Page 107, Equation (7): ``multideg$(a_{ij})$'' should be
``multideg$(a_{ij}g_i)$''
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Page 114, line $-11$: ``Note that different orderings'' should be
``Also observe that different orderings''
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Page 122, statement of Theorem 2: ``factors'' should be
``components''
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Page 125, statement of Corollary 4: ``factors'' should be
``components''
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Page 126, line $-3$: ``At the end of Chapter 2'' should be
``In \S8 of Chapter 2''
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Page 129, line $-6$: ``parameterization'' should be
``parametrization''
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Page 130, third line after (6): ``$g_1\cdots g_n$'' should be
``$g_1g_2\cdots g_n$''
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Page 136, line 2: ``${\bf V}(g)$'' should be ``${\bf V}(f)$''
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Page 144, line 1: ``\S7 of Chapter 2'' should be ``\S8 of Chapter 2''
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Page 145, line 3: ``the $t$ values of where'' should be ``the values
of $t$ where''
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Page 148, line 9: ``$f$ and $g$'' should be ``$g$ and $h$''
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Page 149, line 19: ``Proposition 1'' should be ``Proposition 2''
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Page 151, lines $-10$ and $-11$: ``contradicts (iii)'' should be
``contradicts (ii)''
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Page 153, line $-4$: ``eliminate $y$'' should be ``eliminate
$x$''
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Page 156, first line and second to last line of Exercise 6: ``Exercise
13'' should be ``Exercise 14''
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Page 157, Hint to part a of Exercise 12: ``$\widetilde{A}$ is constant
and $\widetilde{B}$ is linear'' should be ``$\widetilde{A}$ is linear
and $\widetilde{B}$ is constant''
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Page 157, part b of Exercise 13: ``Propositions 6 and 7''
should be ``Propositions 8 and 9''
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Page 157, hint to Exercise 14: ``rows'' should be
``columns''
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Page 157, line $-1$: ``$l \le m$'' should be ``$l \ge m$''
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Page 162, line 12: ``$u_2^{\alpha_1} \cdots u_s^{\alpha_s}$'' should
be ``$u_2^{\alpha_2} \cdots u_s^{\alpha_s}$''
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Page 163, lines $-3$ and $-1$: ``$h(x_1,{\bf
c},u_2,\dots,u_s)$'' should be ``$h({\bf c},u_2,\dots,u_s)$''
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Page 164, lines $4$ and $7$: ``$h(x_1,{\bf
c},u_2,\dots,u_s)$'' should be ``$h({\bf c},u_2,\dots,u_s)$''
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Page 164, line 5: ``equation (4)'' should be ``equation
(3)''
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Page 164, line 12: ``combine (9) and (10)'' should be
``combine (8) and (10)''
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Page 164, line $-7$: ``$f_2 + x_1^Nf_2$'' should be ``$f_2+x_1^Nf_1$''
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Page 165, Exercise 1: ``$g = x^3y-4y^2+y+x^3-4y+1$'' should be ``$g =
x^3y+x^3-4y^2-3y+1$''
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Page 166, part a of Exercise 6: ``$a_0(x_2,\dots,x_i,{\bf c}) \ne 0$
and $a_0(x_2,\dots,x_i,{\bf c}) \ne 0$'' should be
``$a_0(x_2,\dots,x_i,{\bf c}) \ne 0$ and $b_0(x_2,\dots,x_i,{\bf c})
\ne 0$''
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Page 166, part b of Exercise 6: ``used the proof'' should be ``used in
the proof''
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Page 166, first sentence of Exercise 7: ``what happens when to a''
should be ``what happens to a''
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Page 168, line $-5$: ``Proposition 11 of Chapter 2, \S4'' should be
``Proposition 9 of Chapter 2, \S5''
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Page 170, line $-5$: ``factors'' should be ``components''
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Page 171, display near top of page: ``$f_k = 0$'' should be ``$f_s =
0$''
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Page 178, line $-18$: ``dicussion'' should be ``discussion''
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Page 178, line $-15$: ``\S6 of Chapter 3'' should be ``\S5
of Chapter 3''
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Page 178, line $-7$: ``Theorem 2 of Chapter 3, \S6'' should
be ``Theorem 5 of Chapter 3, \S5''
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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 180, line 18: ``use to the product rule'' should be ``use the
product rule''
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Page 181, Exercise 10: ``$\langle h\rangle \supset J$'' should be
``$\langle h \rangle \subset J$''
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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 186, line $-5$: ``On other hand'' should be ``On the other hand''
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Page 189, line 3 of the proof of Theorem 15: ``$x \in
V(I\cap J)$'' should be ``$x \in {\bf V}(I\cap J)$''
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Page 189, line $-1$: Delete ``of $f$ and $g$'' at the end of
the sentence.
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Page 190, second line of Exercise 13: ``Exercise 4 of \S2'' should be
``Exercise 9 of \S1''
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Page 191, line 1: ``$f \in k[x_1,\dots,x_n]$'' should be ``$f \in I$''
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Pages 191--193: ``algebraic variety'' should be ``variety'',
and similarly ``algebraic vareities'' should be ``varieties''
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Page 192, statement of Theorem 3: ``factors'' should be
``components''
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Page 195, line 6: ``$\langle h_1,\dots,h_p\rangle \in I\cap
\langle g\rangle$'' should be ``$\langle h_1,\dots,h_p\rangle = I\cap
\langle g\rangle$''
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Page 195, display in Exercise 7: ``$\dots,x_n]$'' and ``$\dots,y_m]$''
should be ``$\dots,x_m]$'' and ``$\dots,y_n]$''
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Page 198, line 8: ``$\dots f_n(t_1,\dots t_m)$'' should be ``$\dots,
f_n(t_1,\dots,t_m)$''
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Page 198, line $-7$: ``$F(t_1,\dots t_m)$'', ``$g_1(t_1,\dots t_m)$''
and ``$f_n(t_1,\dots t_m)$'' should be ``$F(t_1,\dots,t_m)$'',
``$g_1(t_1,\dots,t_m)$'' and ``$f_n(t_1,\dots,t_m)$''
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Page 198, line $-1$: ``$g_1(t_1,\dots t_m)$'' should be
``$g_1(t_1,\dots,t_m)$''
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Page 199, line 4: ``$k[t_1,\dots t_m]$'' should be ``$k[t_1,\dots,t_m]$''
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Page 199, line 5: ``is the zero'' should be ``is zero''
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Page 199, line 15: ``a irreducible'' should be ``an irreducible''
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Page 199, line $-10$: ``$x_1-a_1,\dots x_n-a_n$'' should be
``$x_1-a_1,\dots,x_n-a_n$''
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Page 200, line 16: ``$x_1-a_1,\dots x_n-a_n$'' should be
``$x_1-a_1,\dots,x_n-a_n$''
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Page 201, line 5: ``maximum ideal'' should be ``maximal
ideal''
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Page 202, line $-1$: ``below'' should be ``on the next
page''
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Page 208, line $-12$: ``Theorem 2 of \S2'' should be ``Theorem 2 of
\S6''
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Page 209, line 9: ``Noether--Lasker'' should be
``Lasker--Noether''
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Page 209, line 20: ``minimal decomposition'' should be ``minimal
primary decomposition''
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Page 211, Exercise 10: ``Propostion 16'' should be
``Proposition 16''
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Page 214, line $-15$: ``$(x_2,x_3)$'' should be ``$(y,z)$''
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Page 227, Exercise 12: ``the to mappings'' should be ``the two
mappings''
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Page 239, line $-9$: ``$\alpha\circ\phi$'' should be
``$\phi\circ\alpha$''
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Page 240, line $-12$: ``some polyonomial'' should be ``some
polynomial''
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Page 247, line 11: ``$y/x^2$'' should be ``$x/y^2$''
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Page 247, line 18: ``$f(y/x^2)$'' should be ``$f(x/y^2)$''
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Page 247, line $-9$: ``$a((y/x^2)^2) + (y/x^2)^5b((y/x^2)^2)$'' should
be ``$a((x/y^2)^2) + (x/y^2)^5b((x/y^2)^2)$''
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Page 247, line $-8$: ``$a(y^2/x^4) + (y^5/x^{10})b(y^2/x^4)$'' should
be ``$a(x^2/y^4) + (x^5/y^{10})b(x^2/y^4)$''
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Page 247, line $-7$: ``$y^2 = x^5$, so $y^2/x^4 = x$ and $y^5/x^{10} =
yx^{10}/x^{10} = y$'' should be ``$x^2 = y^5$, so $x^2/y^4 = y$ and
$x^5/y^{10} = xy^{10}/y^{10} = x$''
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Page 248, line 11: ``From (2), if'' should be ``From (2), it''
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Page 254, part a of Exercise 13: ``$\psi\circ\psi$'' should be
``$\psi\circ\phi$''
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Page 301, part ii of Theorem 4: ``${\bf V}(d_1,\dots,d_n)$'' should be
``${\bf V}(d_1 d_2 \cdots d_n)$''
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Page 306, line 1: ``affect'' should be ``effect''
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Page 311, line 3: ``down Groebner'' should be ``down a Groebner''
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Page 322, part c of Exercise 4: ``$M_{\tau\nu}$, where $\tau\nu$ is
the permutation taking $i$ to $\tau(\nu(i))$'' should be
``$M_{\nu\tau}$, where $\nu\tau$ is the permutation taking $i$ to
$\nu(\tau(i))$''
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Page 323, part d of Exercise 6: ``$I_2$'' should be ``$I_3$''
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Page 341, line $-14$: ``whereas $V_F$ depends'' should be ``whereas
$I_F$ depends''
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Page 346, line 3: ``that the we'' should be ``that we''
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Page 352, line $-2$: ``look by like'' should be ``look like''
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Page 358, line 6: ``$8-2^2 = 4 \ne 0$ should be ``$8-4^2 = -8 \ne 0$''
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Page 363, Exercise 3: ``how a lines'' should be ``how lines''
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Page 364, Exercise 11: ``$f(1,x_1,\dots,x_n)$'' should be
``$f(x_1,\dots,x_n)$''
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Page 364, part c of Exercise 15: ``vareities'' should be ``varieties''
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Page 381, line 12: ``by the equations'' should be ``by the equation''
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Page 428, line 1: ``a ideal'' should be ``an ideal''
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Page 432, line $-5$: ``$k[x_1,\dots,x_n]_s$'' should be
``$k[x_0,\dots,x_n]_s$''
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Page 438, parts a and b of Exercise 18: ``${}^a HF_I(s) -
{}^a HF_I(s)$'' should be ``${}^a HF_I(s) - {}^a HF_I(s-1)$''
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Page 448, line 12: ``$n-(d+1)$'' should be ``$n-d$''
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Page 454, line 3: ``$m \ge \dim V$,'' should be ``$m \ge \dim V$.''
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Page 459, line 2: ``$z$-axis'' should be ``$y$-axis''
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Page 464, line 4: ``charactersitic'' should be ``characteristic''
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Page 466, line 1: ``as the nonzero'' should be ``is the nonzero''
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Page 466, line $-9$: ``as defined in Definition 1'' should be ``as
defined in Definition 1 of the previous section''
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Page 488, line 19: ``$x^2+2y^2$'' should be ``$x^2+3y^2$''
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Page 492, line 15: ``$x^2+2y^2$'' should be ``$x^2+3y^2$''
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Page 495, line $-2$: ``such a Maple'' should be ``such as
Maple''
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Page 500, line $-15$: ``C.~Hoffman'' should be
``C.~Hoffmann''
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Page 504, index entry for {\it curve, twisted cubic\/}: ``363'' should
be ``362'', and ``375, 376, 378'' should be ``366, 375, 376, 377f''
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\end