You have full access to this open access chapter, Download chapter PDF
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 − 3z” 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
so that \(\overline{f}^{G} = -x + y\neq 0\). Yet we can also write
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
Page 160, part (b) of Exercise 4: “go = g3” should be “go = 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, …, xn)). It follows that”
Page 169, line − 7: “u1(x1)” should be “u(x1)”
Page 170, line 1: “〈fi, f∗〉” should be “〈fi, 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 fi(a1, …, an) = 0” should be “must have fi(a1, …, an) ≠ 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 JsM.”
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(Il)”
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(Il)∖”
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:
Page 228: Add the following new exercise:
-
15.
In the setting of Theorem 2, prove that xγ > 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: “VW(a2 − b2 + 4)” should be “VW(y2 − z2 + 4)”
Page 272, second paragraph of the proof of Proposition 6: In two places, “V(figi′ − f′igi)” should be “VV(figi′ − f′igi)”
Page 279, display (1): Replace the display with
Page 279, line − 10: “aiℓsiℓ” should be “aiℓsℓ”
Page 279, line − 7: “the coefficient of x is” should be “the coefficient of xℓ is”
Page 279, the last display should be:
Page 279, line − 4: “ai ∈ R” should be “ci ∈ 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, …, ym]”
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, …, xm, t1, …, tm]” should be “k[x1, …, xn, t1, …, tm]”
Page 309, display in the middle of the page: Replace the display with
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: “Bij ∈ k(t)[x]” should be “Bji ∈ 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 tj 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 Fi and Gj in k[t], we get
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 gj = ∑i = 1sBji fi in k(t)[x] and let dj ∈ k[t] be a polynomial that clears the denominators of Bj1, …, Bjs. Also let F = lcm(F1, …, Fs). 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:
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: “hj−i(xk, …, xn)” should be“hj−i(xj, …, xn)”
Page 355, line 1 of the display in Exercise 15: “ + (−1)j−1σk−1xi + (−1)jσk = ” should be “ + (−1)j−1σj−1xi + (−1)jσj = ”
Page 355, display of Exercise 18: “ = sj = σ1sj−1 +” should be “ = sj −σ1sj−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:
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: ⋯: an)” should be “ϕ(a1, …, an) = (1: a1: ⋯: an)”
Page 404, line 5: “1 ≤ i1 < ” should be “0 ≤ i1 < ”
Page 404, line 2 of Exercise 9: “fi ∈ k[x0, …, xn]” should be “fj ∈ k[x0, …, xn]”
Page 410, line − 1: “k[x1, …, xn]” should be “k[x0, …, xn]”
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(fh)” should be “\(\text{LM}_{>_{h}}(f^{h})\)”
Page 416, line − 4: “lm> h(fh)” should be “\(\text{LM}_{>_{h}}(f^{h})\)”
Page 417, equation (2): “lm> h(fh)” should be “\(\text{LM}_{>_{h}}(f^{h})\)”
Page 417, two lines below equation (2): “lm> h(gih)” 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 Ia(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 ∈ In(0) ∩⋯ ∩ In(n)”
Page 430, line 17: “Now suppose f ∈ I(i)” should be “Now suppose f ∈ In(i)”
Page 430, line 20: “f ∈ I(0) ∩⋯ ∩ I(n)” should be “f ∈ In(0) ∩⋯ ∩ In(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: “wij = λw′ij” should be “w′ij = λwij”
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 = 0naijxixi” should be “∑i, j = 0naijxixj”
Page 448, line 1 of Exercise 9: “be nonzero” should be “be nonzero with Q = (aij) 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 = b0zm + ⋯ ” should be “f = a0zm + ⋯ ”
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: “aHFI(s) =aHPI(s)” should be “aHFR∕I(s) =aHPR∕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, …, fs ∈ k[x1, …, xn]” should be “If f, f1, …, fs ∈ k[x1, …, xn]”
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: “HFS∕〈lt(G)〉(m′) < HFS∕I(m′)” should be “HFS∕〈lt(G)〉(m′) > HFS∕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: “fi ∈ R” should be “fi ∈ R, fi 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”
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Cox, D.A., Little, J., O’Shea, D. (2018). Correction to: Ideals, Varieties, and Algorithms. In: Ideals, Varieties, and Algorithms. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-16721-3_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-16721-3_11
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-16720-6
Online ISBN: 978-3-319-16721-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)