Correction to: Ideals, Varieties, and Algorithms

Correction to: D.A. Cox et al., Ideals, Varieties, and Algorithms, Undergraduate Texts in Mathematics, https://doi.org/10.1007/978-3-319-16721-3

After initial publication of the book, various errors were identified that needed correction. The following corrections have been updated within the current version, along with all known typographical errors.

Chapter 1

Page 23, part (c) of Exercise 5: “Adapt the argument given at the end of the section” should be “Adapt the argument used for the circle x² + y² = 1”

Page 24, line 1: “adapt the argument given at the end of the section” should be “adapt the argument used for the circle x² + y² = 1”

Page 39, line 6 of the paragraph beginning “To see why this algorithm works”: “By (5)”, should be “By (1),”

Page 47, part (a) of Exercise 14: “where h(a) ≠ 0” should be “where r ≥ 1 and h(a) ≠ 0”

Chapter 2

Page 63, bottom display, second line underneath)______: “xy² − x” should be “x²y − x”

Page 63, bottom display, fourth line underneath)______: “x²y − y” should be “xy² − y”

Page 66, lines − 5 and − 6: “[using condition (ii) of the definition of a monomial order]” should be “[using Lemma 8 of §2]”

Page 81, line 1 of Exercise 2: “lt(I)” should be “〈lt(I)〉”

Page 82, last line of Exercise 13: “Exercise 14 of Chapter 1, §4” should be “Proposition 8 of Chapter 1, §4”

Page 89, part (d) of Exercise 5: “z² − 3^z” should be “z² − 3z”

Page 93, line − 3: “〈lt(G∖{p})〉” should be “〈lt(G∖{g})〉”

Page 96, line 2 of part (a) of Exercise 12: “is not divisible by” should be “has leading term not divisible by”

Page 97, line 3 of Exercise 14: “$\frac{x_{j} - a_{j}} {a_{i} - a_{j}}$” should be “$\frac{x - a_{j}} {a_{i} - a_{j}}$”

Page 105, lines 13–16: Replace xy + 1 with xy − 1 in three places and − x − y with − x + y in two places. Thus the lines should be as follows: ______________________

§3. If we divide f = xy² − x by G = (xy − 1, y² − 1), the division algortihm gives

$$\displaystyle{xy^{2} - x = y \cdot (xy - 1) + 0 \cdot (y^{2} - 1) + (-x + y)}$$

so that $\overline{f}^{G} = -x + y\neq 0$. Yet we can also write

$$\displaystyle{xy^{2} - x = 0 \cdot (xy - 1) + x \cdot (y^{2} - 1),}$$

Page 107, line − 4: “f₃ = xz + y − z + 1” should be “f₃ = xz − x + y + 1”

Page 108, Exercise 1: “f₃ = xz + y − z + 1” should be “f₃ = xz − x + y + 1”

Page 110, Lemma 4: The statement of the lemma should be changed to the following:

Lemma  4 . Every element of S(F) can be written as a sum of homogeneous elements of S(F). Furthermore, this decomposition is unique.

Page 115, line 12 “by Lemma 2” should be “by Lemma 2 of §9”

Chapter 3

Page 136, line 20: “$(t,u,x,y,z) \in \mathbf{V}(I) \subseteq \mathbb{R}^{5}$” should be “$(t,u,x,y,z) \in \mathbf{V}(I) \subseteq \mathbb{C}^{5}$”

Page 140, Exercise 3: “t² is always positive” should be “t² is always ≥ 0”

Page 154, part (c) of Exercise 15: The x-coordinate of the second displayed point should be

$$\displaystyle{\pm \frac{1} {2}\sqrt{15 + 6\root{3}\of{2 } - 12\root{3}\of{4}}}$$

Page 160, part (b) of Exercise 4: “g_o = g₃” should be “g_o = g₂”

Page 167, line − 17: Replace “It follows that” with “(Proposition 5 applies to f, g since their coefficients lie in the field k(x₂, …, x_n)). It follows that”

Page 169, line − 7: “u₁(x₁)” should be “u(x₁)”

Page 170, line 1: “〈f_i, f_∗〉” should be “〈f_i, f^∗〉”

Page 170, part (c) of Exercise 3: “part (a) is still true but part (b) can fail” should be “parts (a) and (b) are still true”

Page 171, line 2 of Exercise 9: “l ≥ m” should be “l ≥ m > 0”

Chapter 4

Page 180, line 13: “must have f_i(a₁, …, a_n) = 0” should be “must have f_i(a₁, …, a_n) ≠ 0”

Page 181, lines 1 and 2 of Exercise 10: “$\mathbb{R}[x,y]$” should be “$\mathbb{R}[x,y,z]$” in two places.

Page 181, line 4 of Exercise 10: “same for $\mathbb{R}[x]$” should be “same for $\mathbb{R}[x]$ and $\mathbb{R}[x,y]$”

Page 187, line − 3: “$a_{1} \frac{\partial f_{i}} {\partial x_{j}}h_{i}$” should be “$a_{i}\frac{\partial f_{i}} {\partial x_{j}}h_{i}$”

Page 189, Exercise 15: Replace the hint with “Hint: Show that xy, xz, yz generate the ideal of leading terms of $\sqrt{I}$ and use the definition of Gröbner basis given in Chapter 2, §5.”

Page 196, line 4: “principal ideals is principal)” should be “two principal ideals is principal)”

Page 206. Exercise 5: Replace the hint with “Hint: Examine the generators of J^sM.”

Page 212, Exercise 10: “Theorem 11 implies” should be “Theorem 11 and Proposition 6 of §7 imply”

Page 221, line 1: “by Exercise 3 of Chapter 2, §9” should be “by Exercise 15”

Page 221, Corollary 3: “With the same notation” should be “With k algebraically closed and the same notation”

Page 222, line 4: “for all i” should be “for all such i”

Page 222, line 5 of Proposition 5: “a variety contained in V ” should be “a variety contained in V(I_l)”

Page 223, line − 17: “$W \subsetneq \mathbf{V}(I)$” should be “$W \subsetneq \mathbf{V}(I_{l})$”

Page 223, line − 9: “fails for I, V(I)∖” should be “fails for I, V(I_l)∖”

Page 223, line − 8: “by Proposition 4” should be “by Proposition 4 (we can assume G is reduced)”

Page 224, second display: The display should be as follows:

$$\displaystyle{\mathbf{V}(I_{1})\setminus \mathbf{V}(c_{1}) = \mathbb{C}\setminus \mathbf{V}(y) = \mathbb{C}\setminus \{0\} \subseteq \pi _{1}(\mathbf{V}(I)) \subseteq \mathbf{V}(I_{1}) = \mathbb{C}.}$$

Page 228: Add the following new exercise:

• 15. In the setting of Theorem 2, prove that x^γ > lt(f) implies $\mathbf{x}^{\gamma } > \text{LT}(\bar{f})$ for f ∈ k[x, y].

Page 231, line 9: “EXERCISES FOR §9” should be “EXERCISES FOR §8”

Page 231, part (b) of Exercise 11: “Exercise 4” should be “Exercise 6”

Page 231, Exercise 12: “Use Proposition 9 of §4” should be “Use Exercise 4 of §4”

Chapter 5

Page 247, line 1: “R = k[x, t]” should be “R = k[t]”

Page 247, line 2 of part (b) of Exercise 10: “a, b ∈ k[x]” should be “a, b ∈ k”

Page 256, Exercise 6: “Let $V = \mathbf{V}(x_{3} - x_{1}^{2},x_{4} - x_{1}x_{2},x_{2}x_{4} - x_{1}x_{5},x_{4}^{2} - x_{3}x_{5}) \subseteq \mathbb{C}^{5}$” should be “Let $V = \mathbf{V}(I) \subseteq \mathbb{C}^{5}$ for $I =\langle x_{3} - x_{1}^{2},x_{4} - x_{1}x_{2},x_{2}x_{4} - x_{1}x_{5},x_{4}^{2} - x_{3}x_{5}\rangle \subseteq \mathbb{C}[x_{1},x_{2},x_{3},x_{4},x_{5}]$”

Page 256, line − 1: At the end, add “Assume that the field k is infinite.”

Page 257, part (e) of Exercise 11: “we developed in Chapter 1” should be “we developed in Chapter 1, §2”

Page 260, lines 2–4: These three lines ________________________________

(iii) is proved in the same way as Theorem 11 of Chapter 4, §5. □

When k is algebraically closed, the Weak Nullstellensatz also holds in k[V ]. You will prove this in Exercise 16._____________________________________________________________

should be replaced with the following: ________________________________

(iii) is proved by first showing that the Weak Nullstellensatz also holds in k[V ]. You will prove this in Exercise 16. From here, one proceeds in the same way as Theorem 11 of Chapter 4, §5. □

Page 271, line following second display: “V_W(a² − b² + 4)” should be “V_W(y² − z² + 4)”

Page 272, second paragraph of the proof of Proposition 6: In two places, “V(f_ig_i′ − f′_ig_i)” should be “V_V(f_ig_i′ − f′_ig_i)”

Page 279, display (1): Replace the display with

$$\displaystyle{ s^{\ell} + c_{1}s^{\ell-1} + \cdots + c_{\ell} = 0,\quad c_{ 1},\ldots,c_{\ell} \in R. }$$

(1)

Page 279, line − 10: “a_iℓs_iℓ” should be “a_iℓs_ℓ”

Page 279, line − 7: “the coefficient of x is” should be “the coefficient of x^ℓ is”

Page 279, the last display should be:

$$\displaystyle{\mathrm{det}(A - xI_{\ell}) = (-1)^{\ell}(x^{\ell} + c_{1}x^{\ell-1} + \cdots + c_{\ell}).}$$

Page 279, line − 4: “a_i ∈ R” should be “c_i ∈ R”

Page 280, line 4: “C has entries in R” should be “C has entries in S”

Page 280, line − 13: “(ii) ⇒ (iii)” should be “(i) ⇒ (iii)”

Page 280, line − 9: “divide f by G” should be “divide f by a Gröbner basis G”

Page 282, line − 6: “finite over k[y]” should be “finite over k[y₁, …, y_m]”

Page 288, line 2 of Exercise 6: “means geometrically” should be “means geometrically when k is algebraically closed”

Page 288, line 1 of Exercise 13: “in (4) is” should be “in (5) is”

Page 288, line 2 of part (a) of Exercise 13: “the substitution (5)” should be “the substitution (4)”

Page 289, last line of part (a) of Exercise 17: “ϕ(V(J))” should be “π(V(J))”

Chapter 6

Page 294, line below second display: “C = U × V ” should be “$\mathcal{C} = U \times V$”

Page 300, first display: “f(θ₁ + θ₂ + θ₃)” should be “f(θ₁, θ₂, θ₃)”

Page 302, line 2: “$\mathcal{J} = \mathbf{V}(x_{1}^{2} + y_{1}^{2} - 1,x_{2}^{2} + y_{2}^{2} - 1,x_{3}^{2} + y_{3}^{2} - 1)$” should be

“$\mathcal{J} = \mathbf{V}(c_{1}^{2} + s_{1}^{2} - 1,c_{2}^{2} + s_{2}^{2} - 1,c_{3}^{2} + s_{3}^{2} - 1)$”

Page 302, two lines below (7): “V = V(x₁² + y₁² − 1, x₂² + y₂² − 1, x₃² + y₃² − 1)” should be “V = V(c₁² + s₁² − 1, c₂² + s₂² − 1, c₃² + s₃² − 1)”

Page 304, part (a) of Exercise 9: “result of part (c)” should be “result of part (e)”

Page 305, line 12: “in equation (7) of §2” should be “in equation (6) of §2”

Page 305, line 1 of (2): “$\frac{2bl_{2}l_{3}} {2l_{2}(a^{2} + b^{2})}s_{2}$” should be “$\frac{bl_{2}l_{3}} {l_{2}(a^{2} + b^{2})}s_{2}$”

Page 305, line 2 of (2): “$\frac{2al_{2}l_{3}} {2l_{2}(a^{2} + b^{2})}s_{2}\ +$” should be “$\frac{al_{2}l_{3}} {l_{2}(a^{2} + b^{2})}s_{2}\ -$”

Page 306, line 1 of (3): “$\frac{2b} {2(a^{2} + b^{2})}s_{2}$” should be “$\frac{b} {a^{2} + b^{2}}s_{2}$”

Page 306, line 2 of (3): “$\frac{2a} {2(a^{2} + b^{2})}s_{2}\ +$” should be “$\frac{a} {a^{2} + b^{2}}s_{2}\ -$”

Page 307, line − 3: “when a² + b² < 4,” should be “when 0 < a² + b² < 4,”

Page 308, line 6: “if l₄ lies in” should be “if l₂ = l₃ = 1 and l₄ lies in”

Page 308, line 8 of the subsection Specialization of Gröbner Bases: “k[x₁, …, x_m, t₁, …, t_m]” should be “k[x₁, …, x_n, t₁, …, t_m]”

Page 309, display in the middle of the page: Replace the display with

$$\displaystyle{1,\ l_{2},\ l_{3},\ l_{2},\ l_{3},\ 1,\ l_{2}l_{3},\ l_{2}l_{3},\ l_{2}l_{3},\ a,\ b,\ a^{2} + b^{2},\ l_{ 2}l_{3}.}$$

Page 309, two lines below display: “a, b, l₂, l₃, a² + b² and a² + b² − l₂² − l₃² are nonzero” should be “a, b, l₂, l₃ and a² + b² are nonzero”

Page 312, line 3: “We have” should be “When l₂ = l₃ = 1, we have”

Page 316, line 5: “B_ij ∈ k(t)[x]” should be “B_ji ∈ k(t)[x]”

Page 316, part (c) of Exercise 7: Replace the hint with “Hint: The monomial orders for k(t)[x] and k[x] are the same—the parameters t_j are “constants” as far as the ordering is concerned. Theorem 6 of Chapter 2, §9 will be useful.”

Page 316, lines 7 and 8 of Exercise 8: Delete these lines and replace them with the following: _______________________________________________________

nonzero polynomials F_i and G_j in k[t], we get

$$\displaystyle{\tilde{f}_{i} = F_{i}f_{i},\ \tilde{g}_{j} = G_{j}g_{j} \in k[\mathbf{x},\mathbf{t}].}$$

Page 316, line 9 of Exercise 8: “$\tilde{I} \subseteq k(\mathbf{t})[\mathbf{x}]$” should be “$\tilde{I} \subseteq k[\mathbf{x},\mathbf{t}]$”

Page 316, part (a) of Exercise 8: Replace part (a) with the following: ______

• a. Fix j and suppose g_j = ∑_{i = 1}^sB_ji f_i in k(t)[x] and let d_j ∈ k[t] be a polynomial that clears the denominators of B_j1, …, B_js. Also let F = lcm(F₁, …, F_s). Then prove that

  $$\displaystyle{d_{j} \in (\tilde{I}: F\tilde{g}_{j}) \cap k[\mathbf{t}],}$$

  where $\tilde{I}: F\tilde{g}_{j}$ the ideal quotient as defined in §4 of Chapter 4.

________________________________

Page 316, part (b) of Exercise 8: “$(\tilde{I}:\tilde{ g}_{j}) \cap k[\mathbf{t}]$” should be “$(\tilde{I}: F\tilde{g}_{j}) \cap k[\mathbf{t}]$”

Page 317, part (b) of Exercise 11: “$c_{i} \in \mathbb{R}[a,b,l_{2},l_{3}]$” should be “$h_{i} \in \mathbb{R}[a,b,l_{2},l_{3}]$”

Page 326, line 4 of the first display should be:

$$\displaystyle{f_{4} = x_{3}u_{3} + x_{4}u_{1} - x_{4}u_{2} - u_{1}u_{3},}$$

Page 331, part (b) of Exercise 2: Replace with “b. With this choice, explain why we can specify the coordinates of B as B = (u₃, 0), i.e., the x-coordinate of B is arbitrary, but the y-coordinate is zero.”

Page 332, Exercise 10: “made in Example 1” should be “made in the continuation of Example 1”

Page 332, line − 1: “reducible components” should be “irreducible components”

Page 333, line 2 of part (e) of Exercise 14: “follows from part (a)” should be “follows from part (b)”

Page 333, line 2 of part (e) of Exercise 14: “(c ⋅ g)’s” should be “(c ⋅ g)^s”

Page 333, line 3 of part (c) of Exercise 15: “show that $\overline{c}$ has” should be “show that $c\overline{c}$ has”

Page 336, lines 6-7: Interchange the order of these two lines. Thus the line “q: = ⋯ ” should be above the lines “r: = ⋯ ”

Chapter 7

Page 348, line − 6: “lt(σ₁σ)” should be “lt(σ₁σ₂)”

Page 354, first display of Exercise 11: “h_j−i(x_k, …, x_n)” should be“h_j−i(x_j, …, x_n)”

Page 355, line 1 of the display in Exercise 15: “ + (−1)^j−1σ_k−1x_i + (−1)^jσ_k​ = ” should be “ + (−1)^j−1σ_j−1x_i + (−1)^jσ_j = ”

Page 355, display of Exercise 18: “ = s_j = σ₁s_j−1 +” should be “ = s_j −σ₁s_j−1 +”

Page 356, line 7: “every linear map” should be “every invertible linear map”

Page 361, second display of Example 13: Replace with the following:

$$\displaystyle{x^{i}y^{j} = \left \{\begin{array}{@{}l@{\quad }l@{}} x^{2m}y^{2l} = (x^{2})^{m}(y^{2})^{l} \quad &\text{if i,j are even} \\ x^{2m+1}y^{2l+1} = (x^{2})^{m}(y^{2})^{l}xy\quad &\text{if i,j are odd}. \end{array} \right.}$$

Page 362, Exercise 6: “k[x, y, z]^G” should be “$\mathbb{R}[x,y,z]^{G}$” in part (d) and again in part (e)

Page 362, part (a) of Exercise 7: “k[x, y, z]^G” should be “$\mathbb{R}[x,y,z]^{G}$”

Page 363, second line of the first display: The third factor of g should be “(x − y + z)”

Page 363, part (b) of Exercise 7: “k[x, y, z]^G” should be “$\mathbb{R}[x,y,z]^{G}$” twice one line below the display, once two lines below the display, and once three lines below the display

Page 364, part (b) of Exercise 14: “Use the method of Exercise 13” should be “Use the method of Exercise 12”

Page 367, line − 1: At the end of the display, “$\sum _{\vert \beta \vert =\vert G\vert }R_{G}(x^{\beta })u^{\beta }$” should be“$\sum _{\vert \beta \vert =\vert G\vert }b_{\beta }R_{G}(x^{\beta })u^{\beta }$”

Page 372, part (b) of Exercise 8: “use Exercise 6 and §2” should be “use Exercise 6 and Example 13 of §2”

Page 382, part (c) of Exercise 2: “Use Exercise 13” should be “Use Exercise 16”

Page 382, line 2 of Exercise 9: “b = A ⋅ a” should be “b = A ⋅ a”

Page 383, line 2 of Exercise 12: “G ⋅ b ∪ G ⋅ a −{ a}” should be ‘(G ⋅ b ∪ G ⋅ a)∖{a}”

Page 383, line 2 of Exercise 16: “as in Definition 1 of” should be “as in Definition 2 of”

Chapter 8

Page 394, Exercise 7: “the map (2)” should be “the map (1)”

Page 399, line − 4: “ψ” should be “ϕ” in two places

Page 399, line − 2: “(1: a₁: ⋯: a_n)” should be “ϕ(a₁, …, a_n) = (1: a₁: ⋯: a_n)”

Page 404, line 5: “1 ≤ i₁ < ” should be “0 ≤ i₁ < ”

Page 404, line 2 of Exercise 9: “f_i ∈ k[x₀, …, x_n]” should be “f_j ∈ k[x₀, …, x_n]”

Page 410, line − 1: “k[x₁, …, x_n]” should be “k[x₀, …, x_n]”

Page 414, part (b) of Exercise 13: “V ∖V ∩ V(g)” should be “V ∖(V ∩ V(g))”

Page 416, line 2 of Lemma 5: “lm_{> h}(f^h)” should be “$\text{LM}_{>_{h}}(f^{h})$”

Page 416, line − 4: “lm_{> h}(f^h)” should be “$\text{LM}_{>_{h}}(f^{h})$”

Page 417, equation (2): “lm_{> h}(f^h)” should be “$\text{LM}_{>_{h}}(f^{h})$”

Page 417, two lines below equation (2): “lm_{> h}(g_i^h)” should be “$\text{LM}_{>_{h}}(g_{i}^{h})$”

Page 419, second sentence of the proof of Theorem 8: The sentence should be “Applying the proof of part (i) of Proposition 7 with I in place of I_a(W) shows that Z is a projective variety containing W.”

Page 425, first line following fourth display: “trivial solutions (0: 0; y)” should be “trivial solutions (0, 0, y)”

Page 429, line − 1: “This proves f ∈ I⁽⁰⁾ ∩⋯ ∩ I⁽ⁿ⁾” should be “This proves f ∈ I_n⁽⁰⁾ ∩⋯ ∩ I_n⁽ⁿ⁾”

Page 430, line 17: “Now suppose f ∈ I⁽ⁱ⁾” should be “Now suppose f ∈ I_n⁽ⁱ⁾”

Page 430, line 20: “f ∈ I⁽⁰⁾ ∩⋯ ∩ I⁽ⁿ⁾” should be “f ∈ I_n⁽⁰⁾ ∩⋯ ∩ I_n⁽ⁿ⁾”

Page 431, line 2 of the proof of Proposition 8: “Then the proof of Proposition 7” should be ‘Then Proposition 7”

Page 432, line − 1: “point in $\mathbb{P}^{n}$” should be “point in $\mathbb{P}^{m}$”

Page 433, line − 12: “all have weight d” should be “are all weighted homogeneous of weight d”

Page 443, line below display (9): “σ suppose that” should be “suppose that”

Page 445, first display: “$\left (\begin{array}{*{10}c} a_{0} & a_{1} & a_{2} & a_{3} \\ b_{0} & b_{1} & b_{1} & b_{3} \end{array} \right)$” should be “$\left (\begin{array}{*{10}c} a_{0} & a_{1} & a_{2} & a_{3} \\ b_{0} & b_{1} & b_{2} & b_{3} \end{array} \right)$”

Page 446, line − 3: “w_ij = λw′_ij” should be “w′_ij = λw_ij”

Page 447, line 5: “through two points” should be “through two distinct points”

Page 447, line 10: “are nonzero, and, hence, determine a line L” should be “are nonzero and distinct, and, hence, determine a unique line L”

Page 448, part (a) of Exercise 5: “∑_{i, j = 0}ⁿa_ijx_ix_i” should be “∑_{i, j = 0}ⁿa_ijx_ix_j”

Page 448, line 1 of Exercise 9: “be nonzero” should be “be nonzero with Q = (a_ij) symmetric”

Page 449, line 3 of Exercise 10: “set of all lines” should be “union of all projective lines”

Page 449, part (a) of Exercise 13: At the end of line 2, add “The image of F is called a projective line in $\mathbb{P}^{n}$.”

Page 450, line 5: “$V \subseteq \mathbb{P}^{4}$” should be “$V \subseteq \mathbb{P}^{9}$”

Page 453, line − 6: “nonzero” should be “nonconstant”

Page 456, line 7: “f = b₀z^m + ⋯ ” should be “f = a₀z^m + ⋯ ”

Page 456, line 8: “$b_{0} \in \mathbb{C}\setminus \{0\}$” should be “$a_{0} \in \mathbb{C}\setminus \{0\}$”

Page 464, part (a) of Exercise 5: “nonzero polynomial” should be “nonconstant polynomial”

Chapter 9

Page 472, line 4 of part (b) of Exercise 4: Add “(This is a challenging exercise.)”

Page 481, lines − 7 and − 6: “It is easy to generalize this argument and show” should be “By using the discussion following Lemma 4, one can show”

Page 496, Exercise 7: “^aHF_I(s) =^aHP_I(s)” should be “^aHF_R∕I(s) =^aHP_R∕I(s)”

Page 496, part (c) of Exercise 10: “with Theorem 15 of Chapter 4, §3” should be “with Proposition 1 of §1”

Page 497, part (c) of Exercise 13: “Lemma 5 of §2” should be “Lemma 4 of §2”

Page 506, part (a) of Exercise 14: “part (a) of the proposition” should be “part (i) of the proposition”

Page 506, part (b) of Exercise 14: “part (b) of the proposition” should be “part (ii) of the proposition”

Page 514, part (a) of Exercise 10: “If f₁, …, f_s ∈ k[x₁, …, x_n]” should be “If f, f₁, …, f_s ∈ k[x₁, …, x_n]”

Page 535, line − 1: “k ≥ N” should be “i ≥ N”

Page 537, line 4 of part (d) of Exercise 14: “a curve L ⊆ Γ” should be “a curve $\widetilde{L} \subseteq \varGamma$”

Page 537, hint to part (b) of Exercise 15: “BL₀V ” should be “Bl₀V ”

Page 537, part (b) of Exercise 16: “g(q, tq) = 0” should be “g(tq, q) = 0”

Page 537, part (c) of Exercise 16: “g(q, tq) = 0” should be “g(tq, q) = 0”

Chapter 1 0

Page 549, part (a) of Exercise 7: “matrix M₃ in (3)” should be “matrix M₃ in (5)”

Page 549, line 2 of part (e) of Exercise 7: The line should be “x > y > z and explain its relation to the matrix N₂ in Example 8.”

Page 559, line 21: “HF_{S∕〈lt(G)〉}(m′) < HF_S∕I(m′)” should be “HF_{S∕〈lt(G)〉}(m′) > HF_S∕I(m′)”

Page 564, line 3: “ − v² + ξ² −ζ²” should be “v² −ξ² + ζ²”

Page 564, line 10: “ − uη² + vζ²” should be “uη² − vζ²”

Page 564, line 12: “ − uξ² + uζ² + vη²” should be “uξ² − uζ² − vη²”

Page 564, fourth display: “ −ξ²ζ² + η⁴ + ζ⁴” should be “ξ²ζ² −η⁴ −ζ⁴”

Page 564, sixth display: “ − xz + y² + z²” should be “xz − y² − z²”

Page 570, first display: On the left, “$\frac{\mathrm{lcm}(\text{LM}(f_{i}),\text{LM}(f_{j})} {\text{LT}(f_{j})}$” should be“$\frac{\mathrm{lcm}(\text{LM}(f_{i}),\text{LM}(f_{j}))} {\text{LT}(f_{i})}$” (two errors)

Page 578, line 3 of first display: “xy” should be “ − xy”

Page 578, line − 1: in two places, “(xy” should be “(−xy”

Page 579, line before last display: “ − xy + y²” should be “xy + y²”

Page 582, line 2 of Example 6: “f₂ = xy² − xy” should be “f₂ = xy² + xy”

Page 582, first display of Example 6: “ − x(xy² − xy)” should be “ − x(xy² + xy)”

Page 582, line after first display of Example 6: “equal to lt(f₁)” should be “equal to −lt(f₁)”

Page 582, third display: “$\mathfrak{s}(\mathbf{g}) = x\mathbf{e}_{2}$” should be “$\mathfrak{s}(\mathbf{g}) = -x\mathbf{e}_{2}$”

Page 585, line above the second display: “ye₁ − xe₁” should be ‘ye₁ − xe₂”

Page 585, line below the second display: “$\mathfrak{s}(\mathbf{h}) = x^{2}\mathbf{e}_{2}$ divides $\mathfrak{s}(\mathbf{k})$” should be “$\mathfrak{s}(\mathbf{k}) = x^{2}\mathbf{e}_{2}$ divides $\mathfrak{s}(\mathbf{h})$”

Page 586, first line of pseudocode: “f_i ∈ R” should be “f_i ∈ R, f_i monic”

Page 590, part (c) of Exercise 1: On the first line, “order on S” should be “order on R”

Page 590, part (c) of Exercise 1: On the second line, “ > ” should be “ >_POT”

Page 590, part (b) of Exercise 2: Replace with “If we allow $\mathfrak{s}$-reductions and reduction by the syzygy h = (−f₂, f₁), then show that we can reduce g to (0, 0).”

Page 590, part (c) of Exercise 2: Replace with “Use Propositions 12 and 14 to explain why the computations in parts (a) and (b) are unnecessary.”

Page 591, line 10 of the pseudocode in Exercise 5: “ < ” should be “ <_POT”