IDEALS, VARIETIES AND ALGORITHMS
ERRORS CORRECTED IN THE SECOND PRINTING
p.viii: Line -15: Should be "New England Consortium for" rather than
"New England Consortium of"
p.9: Line 9: Should be "n unknowns" rather than "n unknown"
p.16: Line 6: Should be "precise definition" rather than "precise
defintion"
p.24: Exercise 9: Should be "mathematicians" rather than
"mathetmaticians"
p.24: Exercise 9: Should be "Isaac Barrow" rather than "Issac Barrow"
p.28: Part d of Exercise 16: Should be "control polygon" at end of first
sentence rather than "contol polygon"
p.36: Exercise 10: Should be "discussion" rather than "dicussion"
p.41: Part (iii) of Proposition 6: Should be "an algorithm" rather
than "a algorithm"
p.41: Second sentence of third paragraph of Proof: Should be "ability
to find" rather than "ability find"
p.43: Part (iv) of Proposition 8: Should be "an algorithm" rather than
"a algorithm"
p.46: Exercise 13: Should be "exercise" rather than "exericse"
p.47: Second sentence of Exercise 16: Should be "an algorithm" rather
than "a algorithm"
p.79: Second line from the bottom: the subscript should be "ji" rather
than "ij"
p.80: On the last line of Exercise 13, it should read "Use Exercise 14
of" rather than "Use Exercise 13 of"
p.81: In Proposition 1, should be "$k[x_1,\dots,x_n]$ and let $f \in
k[x_1,\dots,x_n]$" rather than "$k[x_1,\dots,x_\alpha]$ and let $f \in
I". Also, on the first line of Proposition 1, the $k[x_1,\dots,x_n]$
should not be broken across two lines
p.81: Delete the word "in" in line 11. Thus "using in the division
algorithm" should be "using the division algorithm"
p.81: On the 12th line from the bottom, "If the remainder zero" should
be "If the remainder is zero"
p.85: In the displayed equation (8), "b_{ijk}" should be "b_{ijk}g_i"
p.89: At the top of the page, replace the two sentences "Both the
basic idea and many refinements that improve the efficiency of the
computations were developed by Buchberger and his coworkers in the
1970s and 1980s. We will consider some of these refinements in \S9"
by "This algorithm of Buchberger is the cornerstone of computational
algebraic geometry. In the 1970s and 1980s, Buchberger and his
coworkers made many refinements to improve the efficiency of the
algorithm. We will consider this topic in \S9."
p.90: On line 16, it should be "" rather than "LT(G - {p})"
p.94: In Example 1, the fourth member of the displayed Groebner basis should
be "xy^4-z^4" rather than "xy^4-z^3"
p.124: On line 7 from the bottom, it should be "the above" rather than
"the the above"
p.125: In the second paragraph, there is a blank line between the
words "simultaneously" and "vanish". This blank line should be removed
p.128: On line 12, it should be "strategy" rather than "stategy"
p.128: On the 16th line from the bottom, it should be "variety in k^n$"
rather than "variety in k^m"
p.131: On line 6, it should be "x_n" at the end of the line rather
than "x_m"
p.131: On line -15 (which is the third line below the commutative
diagram), it should be "x_n" rather than "x_m"
p.134: In the second line of part b of Exercise 12, replace "x^2y-z^3)" with
"x^2y-z^3,vz-xy)"
p.142: The illustration is a bit incorrect -- the two branches of the
singular point need to cross in the center of the picture. This needs
to be done by someone working with the PostScript file in a program
like PageMaker or Adobe Illustrator
p.151: Should delete period at the end of Part (ii) of Lemma 6
p.153: Second sentence of first paragraph of Proof: Should be "is an
integer" rather than "is a integer"
p.153: On the fifth line from the bottom, it should be "Proposition 8"
rather than "Proposition 4"
p.158: In the displayed formula of part a of Exercise 15, the b_0 should be
preceeded by a (-1)^m
p.158: In the displayed formula of part b of Exercise 15, the b_0 to a power
should be preceeded by a (-1)^{m(l-deg(\tilde f))}
p.158: In the displayed formula of part c of Exercise 15, the b_0 to a power
should be preceeded by a (-1)^{m(l-deg(r))}
p.158: In the second display of Exercise 16, the following changes are needed:
first line: put (-1)^{deg(g)(deg(f)-deg(r))} in front of the power of b_0
second line: in the exponent of (-1), change g, r to f, g respectively
third line: the exponent of (-1) should be deg(f)deg(g)+deg(r)(deg(g)-deg(r'))
fourth line: in the exponent of (-1), change g, r, r, r' to f, g, g, r
p.158: On the last line of the page, in the exponent of (-1), replace r by h
p.169: On line 17, it should be "every nonconstant" rather than "every
nonzero constant"
p.220: On the fifth line from the bottom, it should be "it follows"
rather than "if follows"
p.229: On the fourth line from the bottom, replace "below" with "on
the next page"
p.229: The sentence on the third line from the bottom should begin on
the line above immediately after the preceding sentence
p.233: the second line after the proof of Corollary 7: the first polynomial
on the line should be "xy^3-x^2" instead of "xy^2-x^2"
p.233: in the last display on the page, "(1/2,3/4)" should be "(-1/2,3/4)"
p.237: On the first line of the page, replace "k[x_1,\dots,x_n]" by "k[V]"
p.237: On the second line of the page, replace "k[V]" by "k[x_1,\dots,x_n]"
p.243: Replace part a of Exercise 2 with the following sentences: "Let
$\phi \in \cx[V]$. Show that ${\bf V}_V(\phi) = \emptyset$ if and
only if $\phi$ is invertible in $\cx[V]$ (which means that there is
some $\psi \in \cx[V]$ such that $\phi\psi = [1]$ in $\cx[V]$)."
p.244: In the second sentence of part b of Exercise 7, replace
"x < y < t" by "x > y > t"
p.249: On the fifth line from the bottom, replace "Definition 3" with
"Definition 5"
p.253: On the next to last line of Exercise 10, "represented" should be
"representing"
p.285: In the second line of equation (7), replace "(x_6-x_8)^2" with
"(0-x_8)^2"
pp.295-296: The sentence on line -11 ending "... the following figure" should
say "the figure on the following page". And the next sentence (which
begins "Our theorem ...") should be the start of a new paragraph and
hence should be indented
p.300: In the last line of the display at the bottom of the page, replace
"+x_4^2-2x_4x_8" with "-x_4^2+2x_4x_8"
p.301: In the display at the top of the page, replace "x_2x_3^3" with
"x_2x_3^2"
p.302: In the expression for R7, "x_1 x_4 x_7" should be "2 x_1 x_4 x_7"
p.319: line -2 "ring" should be "a ring"
p.334: Capitalize "suppose" in Part (ii) of Proposition 1
p.334: On the sixth line from the bottom, replace "exericse" with "exercise"
p.347: On the 13th line from the bottom, it should be "third quadrant"
rather than "fourth quadrant"
p.354: In part d of Exercise 5, it should be "a hyperbola" rather
than "at hyperbola"
p.359: In the display on the third line from the bottom, it should be
"g_i(a_1,...,a_n)" rather than "g(a_1,...,a_n)"
p.375: On line 9, it should be "Exercise 8" rather than "Exercise 7"
p.382: Some print is missing in the first two sentences of the last paragraph
-- the words "use" and "Suppose" are partially obscured
p.393: In Part b of Exercise 7, "v^2y(1+y)" should be "v^2y^2(1+y)" in
the first sentence
p.393: In Part b of Exercise 7, replace "whereas $uy(1+y)$ and
$vy(1+y)$ do not" by "whereas $uy(1+y)$, $vy(1+y)$ and $vy^2(1+y)$ do
not"
p.393: In Part c of Exercise 7, replace "" by ""
p.425: In the last line of Part a of Exercise 12, it should be
" ... = a_d = 0" rather than " ... = a_n = 0"
p.484: Second sentence of Section 3: Should be "They are similar"
rather than "They similar"
p.489: The first sentence of the last paragraph of section 1 should say "by
Albert Lin and Philippe Loustaunau of George Mason University", and the
third sentence of the paragraph "It also gives information ... that occur"
should be deleted