Below is the known errata for An Invitation to Analytic Combinatorics: From One to Several Variables by Stephen Melczer – see the textbook website for the textbook manuscript and associated computer algebra worksheets. Many of these errors were found with the help of ChatGPT and Claude (all items were verified by the author). The examples appearing in the associated computer algebra worksheets have been verified to be correct.
Please email any further errors you notice to the author.
The following errata affect results that readers may apply directly, and should be noted by everyone using the text.
Due to editor error and a breakdown in Springer’s publication process the original version of the textbook posted to the publisher website had an incorrect title and author affiliation. This has been fixed as of February 2021 and the correct title should appear in all online and printed copies.
Proposition 2.11 (Page 48): The asymptotic expansion for $f_n$ should be $$ f_n = \omega^{-n}\left( \sum_{k=N}^{N+M} c_k \frac{\Gamma(n-k/R)}{\Gamma(-k/R)\Gamma(n+1)} + O\left(n^{-(N+M+1)/R - 1}\right) \right). $$
Corollary 2.1 (Page 49): The exponents on $\beta$ and $\omega_j$ in the displayed asymptotic formula should be $-n$ instead of $n$. The correct formula is $$ f_n = \frac{\beta^{\color{red}{-n}}n^s}{\Gamma(s+1)} \sum_{j=0}^{m-1}C_j\omega_j^{\color{red}{-n}} + O(\beta^{\color{red}{-n}}n^t). $$
Proposition 2.19 (Pages 68 – 69): The statement that the indicial polynomial has only rational roots should be read to mean that the indicial polynomial at any singularity has only rational roots. The same phrasing also appears in the lead-in paragraph on page 68.
Proposition 2.21 (Page 78): The Cauchy integral formula for derivatives is missing its factor of $n!$ and should read $$ f^{(n)}(w) = \frac{{\color{red}n!}}{2\pi i}\int_\gamma \frac{f(z)}{(z-w)^{n+1}}\,dz. $$
Proposition 3.1 (Page 97): The statement of the Implicit Function Theorem omits the hypothesis ${\color{red}f(\mathbf{w}) = 0}$. The equivalence “$f(\hat{\mathbf{z}}, y) = 0$ if and only if $y = g(\hat{\mathbf{z}})$” should also be qualified to hold for $(\hat{\mathbf{z}}, y)$ in a neighbourhood of $\mathbf{w}$.
Definition 3.9 (Page 101): ${\color{red}\textbf{Incorrect definition of minimal points}}$. The final sentence should read: Equivalently, a minimal point $\mathbf{w}$ is a singularity of $F$ such that no other singularity $\mathbf{z}$ of $F$ satisfies ${\color{red}|z_j| < |w_j| \text{ for each } 1 \leq j \leq d}$. The restatement of this definition at the start of Chapter 5 (Page 185) repeats the original formulation and should be corrected in the same way.
Definition 3.24 (Page 125): The non-negative series extraction operator takes only those terms with non-negative exponents (not positive exponents as written).
Proposition 4.4 and Theorem 4.1 (Page 158): Both displays are missing a factor of $xy$. The series extraction in Proposition 4.4 should be applied to $$\frac{{\color{red}\bar{x}\bar{y}}\sum_{g} \operatorname{sgn}(g)\, g(xy)}{1-tS(x,y)},$$ and the diagonal expression in Theorem 4.1 should read $$ Q(a,b,t) = \Delta\left(\frac{{\color{red}xy}\,O(\bar x, \bar y)}{(1-x)^a(1-y)^b\left(1-txyS(\bar x,\bar y)\right)}\right). $$
Proposition 4.5 (Page 161): Not being contained in a half-space means there is no non-zero $\mathbf{n} \in \mathbb{Z}^d$ such that $\mathbf{i}\cdot\mathbf{n} \geq 0$ for all $\mathbf{i} \in \mathcal{S}$.
Equation (4.12) (Page 167): The final sum enters with the wrong sign and must exclude $V = \emptyset$. The functional equation should end $$ \cdots + t(z_1\cdots z_d)S(\mathbf{z})W(\mathbf{z},t) \; {\color{red}+} \; t\!\!\sum_{{\color{red}\emptyset \neq V \subseteq [d]}}\!\!(-1)^{|V|}(z_1\cdots z_d)S(\mathbf{z})W(\mathbf{z},t)\Big|_{z_j = 0,\, j \in V} .$$ The same incorrect version of this formula is reproduced in Problem 6.2 (Page 261).
Proposition 4.10 (Page 171): The statement should conclude in the orthant $\mathbb{N}^d$ (not $\mathbb{N}^2$).
Theorem 5.1 (Page 186): The final sentence should say $\mathbf{w}$ is minimal if and only if $H(t{\color{red}|}w_1{\color{red}|},\dots,t{\color{red}|}w_d{\color{red}|})\neq0$ for all $0 \leq t < 1$ (moduli bars are missing).
Definition 5.6 (Page 206): The definition applies to a minimal point $\mathbf{w} \in \mathcal{V}\cap{\color{red}\partial}\mathcal{D}$.
Proposition 5.3 (Page 210): The right hand side of Equation (5.22) should read $$ \left(\frac{2\pi}{n}\right)^{k/2}\det(r\mathcal{H})^{-1/2} \left(\sum_{j=0}^M K_j(rn)^{-j} + O\left(n^{-M-1}\right)\right).$$
Lemma 5.4 (Page 211): The conclusion $\Re(r_d\phi(\boldsymbol\theta)) > 0$ must exclude $\boldsymbol\theta = \mathbf{0}$ (where $\phi$ vanishes).
Theorem 5.3 (Page 214): The leading constant in (5.28) is missing a power of $r_d$ and should read $$ \frac{{\color{red}r_d^{p-1}}(2\pi)^{(1-d)/2}}{\sqrt{\det r_d\mathcal{H}}}.$$
Lemma 5.7 (Page 218): The point $\mathbf{w}$ does not need to have non-negative coordinates (which makes the conclusion trivial). The corrected conclusion of the Lemma is: If $\mathbf{w}\in\mathcal{V}_*\cap\partial\mathcal{D}$ is a minimal point of $F(\mathbf{z})$ then the point $|\mathbf{w}|$ with positive coordinates also lies in $\mathcal{V}$ (and is thus also minimal).
Proposition 5.4 (Page 219): For a general Laurent expansion the equation $|\mathbf{z}|=t|\mathbf{w}|+(1-t)|\mathbf{x}|$ should be replaced by $\text{Relog}(\mathbf{z}) = t\text{Relog}(\mathbf{w})+(1-t)\text{Relog}(\mathbf{x})$ (or, equivalently, $|z_j| = |w_j|^t|x_j|^{1-t}$ for all $j$). The statement for power series is correct as written in the book.
Lemma 5.8 (Page 230): The point $\mathbf{z}(\mathbf{r})$ is a smooth critical point near $\color{red}\mathbf{w}$.
Definition of $C_\mathcal{A}$ and $C$ (Pages 251, 256, 257 and 260): The coordinates in the definition of $C_\mathcal{A}$ (in Theorems 6.2 and 6.3) and of $C$ (on pages 251 and 256), should be $$\left(\mathbf{x},{\color{red}\frac{1}{x_1\cdots x_d\,S(\mathbf{x})}}\right)$$ instead of $(\mathbf{x},1/S(\mathbf{x}))$.
Proposition 7.1 (Pages 270 – 271): The hypothesis $\mathbf{a},\mathbf{b} \in \mathbb{R}^d_*$ should read $\mathbf{a},\mathbf{b} \in {\color{red}\mathbb{R}^d}$ (the real and imaginary parts can be zero – they cannot both be zero, but this is already forbidden by $\mathbf{z} \in \mathbb{C}_*^d$).
Algorithm 1 (Page 274): The test in the second line should fail if ${\color{red}H(\mathbf{0}) = 0}$ instead of $H(\mathbf{0}) \neq 0$. Furthermore, the matrix $\tilde{\mathcal{H}}$ should be defined as the polynomial matrix $(z_dH_{z_d})^{\color{red}3}\mathcal{H}$ to clear the denominators in the entries of $\mathcal{H}$ as given in Equation (5.25). The returned quantity $B$ then becomes $(r_dQ_\lambda)^{{\color{red}3(d-1)}}(P^{\prime})^{{\color{red}4-3d}}/Q_{\tilde{\mathcal{H}}}$ to compensate.
Proposition 7.2, Sturm’s Theorem (Page 279): The sign variation difference is reversed: the number of zeroes of $f$ in $(a,b]$ equals ${\color{red}V_f(a) - V_f(b)}$, not $V_f(b) - V_f(a)$.
Corollary 8.1 (Page 320): The assumption should be that $\mathbf{r}$ is generic (not nongeneric).
Lemma 8.2 (Page 324): The height inequality in the final condition is reversed. The conclusion should be that $\nu_{B,\boldsymbol{\sigma}} = 0$ unless $B$ is bounded, $B$ and $\boldsymbol{\sigma}$ lie in the same orthant of $\mathbb{R}^d$, and ${\color{red}h(\boldsymbol{\sigma}) \leq h(\boldsymbol{\sigma}_B)}$.
Proposition 8.4 (Page 333): The displayed equation for $I_\sigma$ should be $$ I_\sigma = {\color{red}\frac{\operatorname{sgn}(\boldsymbol{\sigma})\boldsymbol{\sigma}^{-n\mathbf{r}}}{|\det M|(2\pi)^{d-s}}} \sum_{j=0}^{p_{k_1}+\cdots+p_{k_s}-s} n^j\int_{\mathbb{R}^{d-s}} A_{j,\boldsymbol{\sigma}}e^{-n\phi_\boldsymbol{\sigma}}\,dy.$$
Algorithm 4 (Page 337): The criterion “matrix with rows $b^{(j)}$ not full rank” to run SimpleDecomp is too simplistic. It should test whether any subset of hyperplanes meeting at a common point has linearly independent normals.
Theorem 8.3 (Page 346): The exponential term in the asymptotic expansion should be $\sigma^{{\color{red}-n\mathbf{r}}}$. Furthermore, in the integration equation for $I$ the exponential term should also be $\sigma^{{\color{red}-n\mathbf{r}}}$, the integrand should be $e^{-{\color{red}n}\psi(y)}$ instead of $e^{-\psi(y)}$, and the term $n^{p_1+\cdots+p_d-d}$ should be $n^{\color{red}p_1+\cdots+p_{d-1}-(d-1)}$.
Theorem 9.1 (Page 366): $\Gamma_{\mathbf{w}}$ is the modified log-normal matrix defined on page 365, not the parameterized Hessian matrix from Definition 9.10.
Theorem 9.2, Equation (9.7) (Page 367): The equation as stated has some minor typos, and gives correct sign only for real $\mathbf{w}$. To correct these errors, write $(\nabla_{\log}H_i)(\mathbf{w}) = \tau_i v_i$ with $v_i$ real as in Definition 9.9, and use the leading constant $$ \frac{(2\pi)^{\frac{s-d}{2}}\,G(\mathbf{w})\prod_{i=1}^s\left(-(\mathbf{r}\Gamma_\mathbf{w}^{-1})_i\right)^{p_i-1}}{u(\mathbf{w})\,(\mathbf{p}-\mathbf{1})!\,\prod_{i=1}^s(-\tau_i)\,|\det\tilde{\Gamma}_\mathbf{w}|\,\sqrt{\det Q_\mathbf{w}}} $$ where $\tilde{\Gamma}_\mathbf{w}$ is the matrix with rows $v_1,\dots,v_s,e^{(\pi_1)},\dots,e^{(\pi_{d-s})}$.
Theorem 9.3 (Page 375): The error-term condition $0 < \tau < |\mathbf{w}^{\mathbf{r}}|$ should be $0 < \tau < |\mathbf{w}^{{\color{red}-\mathbf{r}}}|$.
Theorem 10.3 (Page 395): In the first bullet, the $t$-coordinate of the negative-drift contributing points should be $t = 1/({\color{red}w_1\cdots w_{d-1}}\,w_d\bar{S}(\hat{\mathbf{w}}, w_d))$.
The following errata may confuse a reader, but do not change the overall correctness of the main results in the book.
Page xvii (List of Symbols): The entry for $\mathcal{V}(f_1,\dots,f_r)$ should be “Solutions of the system $f_1(\mathbf{z}) = \cdots = f_r(\mathbf{z}) \, {\color{red}=0}$ for $\mathbf{z} \in \mathbb{C}{\color{red}^d}$”.
Section 1.1.2 and Figure 1.2 (Pages 6 and 8): The second step set $\mathcal{S} = \{(0,-1), (\pm1,1) \}$ should be $\mathcal{S} = \{(0,1), (\pm1,-1) \}$ in order for $b_n$ to have the stated asymptotic behaviour. The same incorrect step set also appears in the caption of Figure 1.2 on page 6, and the right panel of Figure 1.2 was generated from the incorrect set.
Pages 7, 65, 73, and 176: The differential equation for the quadrant walk generating function mixes its variables in Equation (1.1), Example 2.23, page 73, and Problem 4.4. The arguments printed as $A^{\prime\prime}(t)$, $A^{\prime}(t)$, and $A(t)$ should be $A^{\prime\prime}(z)$, $A^{\prime}(z)$, and $A(z)$.
Page 7: In $a_n = (1.273\dots)\Psi_1(n) + (5.092\dots)\Psi_2(n)$, the second connection coefficient should be ${\color{red}0.318\dots}$ instead of $5.092\dots$.
Page 9: In the large integral at the top of the page, the factor $(1+v^2)^{1/2}$ in the denominator of the integrand should be ${\color{red}(1+v)^{1/2}}$.
Proof of Proposition 2.3 (Page 31): The displayed integral for $I_n$ should have leading constant ${\color{red}\frac{1}{2\pi i}}$ not $\frac{1}{(2\pi i)^n}$ as written.
Page 47: In both displayed Puiseux series for $zy^3 - y + 1$, the coefficient of $z$ should be ${\color{red}-\tfrac{1}{2}}$ rather than $+\tfrac{1}{2}$.
Example 2.15 (Page 50): The two error terms $O(1/n)$ should be ${\color{red} O(n^{-3/2})}$.
Example 2.16 (Page 52): The expansions ${\color{red}B_1}$ and $B_3$ are singular at $z=\rho$, not $B_2$ and $B_3$ as written.
Proof of Proposition 2.13 (Page 58): The term $F_{\leq k}(z)$ in the equation $$\sum_{n\geq0}f_{n+k}z^n = \frac{F(z) - F_{\leq k}(z)}{z^k}$$ should be ${\color{red}F_{\leq k-1}(z)}$, and the same in the following equation.
Example 2.22 (Pages 63 – 64): The denominator $x^2 - x - t$ should be ${\color{red}x - x^2 - t}$ in three places.
Example 2.23 (Page 65): The solution space is a ${\color{red}\mathbb{C}}$ vector space, not a $\mathbb{C}(z)$ vector space.
Example 2.24 (Page 68): The logarithmic terms in the expansion of $B(z)$ should be ${\color{red}\log\left(\frac{1}{1-16z}\right)}$ (or the connection coefficient for $B$ should be ${\color{red}-1/\pi}$). Furthermore, the final asymptotic result follows from Remark 2.4 not Proposition 2.17.
Example 2.27 (Page 72): The displayed expansions for the $\Gamma_j$ have three typos: the first formula should read $\Gamma_0 = \log(z-1/2)(1 {\color{red}\,-\,} 2(z-1/2) + \cdots)$; the factor $(1-z/2)^3$ appearing in $\Gamma_1, \Gamma_2, \Gamma_3$ should be $\color{red}(z-1/2)^3$; and the coefficient of $(z-1/2)^3$ in $\Gamma_2$ should be ${\color{red}4/25}$ not $8/25$. Furthermore, in the equality $a_0\Psi_0(z)+\cdots+a_3\Psi_3(z) = b_0\Gamma_0(z)+\cdots+b_3\Gamma_3(z)$ the $\Psi_j$ should be $\Phi_j$.
Page 77: The statement “the integral depends only on the image $\gamma([a,b])$, not on the actual function $\gamma$” should say the integral is “invariant under orientation-preserving reparametrization”.
Page 79: In the definition of a pole, $c_n = 0$ should hold for all $n {\;\color{red}<} -M$.
Page 82: The gamma function has poles at the non-positive integers (not the negative integers as stated).
Problem 2.15 (Page 85): In the first bullet, the equality $y = a(z) + b(z)$ should be $\color{red} a(x) + b(x)$. In the first two bullets the second resultant argument $Q(x,y)$ should be $Q(x,{\color{red}z})$. In all three bullets the resultant lies in $\color{red}\mathbb{Q}[x,y]$, not $\mathbb{Q}[y,z]$.
Definition 3.1 (Page 93): The coefficients $f_{\mathbf{i}}$ should be in $\color{red}\mathbb{K}$ instead of $\mathbb{K}^d$.
Page 98: The displayed series should be $\frac{\sin(x-y)}{x-y} = \sum_{n\geq0}\frac{(x-y)^{2n}(-1)^n}{{\color{red}(2n+1)!}}$.
Proof of Proposition 3.2 (Page 98): The extended Euclidean identity is printed with cofactors $b(\hat{\mathbf{z}}), c(\hat{\mathbf{z}})$ depending only on $\hat{\mathbf{z}}$; the cofactors must be polynomials in all variables, $a(\hat{\mathbf{z}}) = b(\mathbf{z})G(\mathbf{z}) + c(\mathbf{z})H(\mathbf{z})$ (only $a$ is free of $z_d$).
Proof of Proposition 3.6 (Pages 101 – 102): The proof is only valid for power series expansions (the statement “Since $w$ is minimal the modulus of $z_d$ cannot decrease when the modulus of $z_j$ decreases” is false for general Laurent expansions). A correct proof of the general case follows from studying the intersection of the tangent spaces to $\mathcal{V}$ and $T(\mathbf{w})$ at $\mathbf{w}$. The proof of Proposition 3.13 on page 119 has the same issue.
Example 3.5 (Page 105): In fact $\rho_+(y) \to {\color{red}1}$ (not $1/2$) as $y \to 0$.
Pages 106 and 136: The statement “every globally bounded function which is analytic at the origin is a G-function” requires D-finiteness. Both the discussion on page 106 and Problem 3.4 should add a D-finite hypothesis.
Example 3.9 (Page 107): The point $\omega = e^{2i/3}$ should be $\omega = e^{2{\color{red}\pi} i/3}$.
Example 3.12 (Page 110) and Example 5.10 (Page 226): The final term in the polynomial $P$ of Example 3.12 should be negated to obtain $P(u,z,y) = z − xy − (x + y + xy)z \, {\color{red} - \; xz^2}$. The rational function in these examples also has a spurious $-z$ in its denominator, which should be $1-xyz-xy-xz-yz-x$ (the critical point $\mathbf{z}_*$ given in Example 5.10 is computed from the corrected denominator, not the printed one).
Footnote 8 (Page 123): The statement “the contour is a subset of the amoeba boundary” should be replaced by “the contour lies in the amoeba but need not lie inside its boundary”.
Page 124: The curve $\Gamma = \operatorname{Relog}(a,b)$ should actually be $\Gamma = {\color{red}\operatorname{Relog}^{-1}((a,b))}$.
Page 132: The algebraic equation $x^2 + y^2 = \epsilon$ for the circle $|z| = \epsilon$ should be $x^2 + y^2 = {\color{red}\epsilon^2}$.
Pages 133 – 134: The statement $f_n = p_j(n)$ in the definition of quasi-polynomial should hold for all $n \equiv j \bmod {\color{red}r}$.
Problem 3.3 (Page 136): The problem should ask to prove Proposition ${\color{red}3.5}$ using the multivariate Cauchy integral formula (not Proposition 3.3).
Problem 3.13 (Page 138): The function $F(z)$ is transcendental over $\color{red}\mathbb{C}(z)$ not $\mathbb{C}[[z]]$.
Equation (4.4) (Page 147): The integral representation for $E(t)$ is missing a leading factor of $\color{red}\frac{1}{2\pi i}$.
Example 4.2 (Page 151): The second singular contribution should be $(-2)^nn^{-3/2}/(2\sqrt{{\color{red}2}\pi})$.
Page 154: The condition that $\mathcal{S}$ contains a step with negative $x$-coordinate, and a step with negative $y$-coordinate, should ask for steps with positive coordinates.
Page 163: The quantity $S_{yy}(a,b) = 2b^{-3}A_{-1}(a)$ should be $2b^{-3}{\color{red}B_{-1}}(a)$.
Example 4.7 (Page 164): The formula for the exponent of $n$ should be $$\alpha = 1 + \pi/\arccos\left({\color{red}-}\frac{1}{2\sqrt{(1+a)(1+b)}}\right).$$
Proof of Proposition 4.10 (Page 171): The step set for $d \geq 3$ should be $\mathcal{S}_d = \mathcal{S}_2 \times \{\pm1\}^{{\color{red}d-2}}$.
Page 199: In the bullet point near the bottom of the page, the equality $a = |x_\ast|$ should be $\color{red}|a| = x_\ast$.
Page 200 (Section 5.1.2): In the final displayed equation, the leading constant $\sqrt{\frac{s}{2r(r+s)\pi n}}$ should be ${\color{red}\sqrt{\frac{r+s}{2rs\pi n}}}$.
Page 200 (Section 5.2): The equality $B = \operatorname{Relog}^{-1}(\mathcal{D})$ should be $\color{red}B = \operatorname{Relog}(\mathcal{D})$. Furthermore, the index set $\mathbb{Z}^n$ should be $\color{red}\mathbb{Z}^d$.
Proof of Corollary 5.1 (Page 207): The difference of integrals in the final displayed equation on Page 207 equals $\color{red}\operatorname{sgn}(r_d)R_1(\hat{\mathbf{z}})$ rather than $-\operatorname{sgn}(r_d)R_1(\hat{\mathbf{z}})$.
Example 5.6 (Page 220): The asymptotic expansion at $\sigma$ has errors in its first and fourth terms. Corrected, the asymptotic contribution of $\sigma$ is $$ \Phi_{\sigma} = 4^n \left( \frac{{\color{red}4}}{\pi n} - \frac{6}{\pi n^2}+\frac{19}{2\pi n^3} - \frac{{\color{red}63}}{4\pi n^4} + O\left(\frac{1}{n^5}\right)\right). $$ The final asymptotic expansion of $f_{n,n,n}$ in the text is correct as presented.
Example 5.7 (Page 222): There is an extra factor of $s$ in the denominator of the leading constant. The leading constant should read $(rn)!(sn)!/\sqrt{2 \pi ae^{-a}(be^{-b}+ae^{-a}-ab)}$.
Proof of Lemma 5.8 (Page 231): The final displayed equation should be $\,\boldsymbol\epsilon^T{\color{red}M^{-1}}\boldsymbol\epsilon/2$ rather than $\boldsymbol\epsilon^TM\boldsymbol\epsilon/2$.
Proof of Proposition 5.10 (Page 233): The bound derived from (5.31) should read $\nu_n(\hat{\mathbf{s}}) \leq e^{-C_3n^{{\color{red}2p-1}}}$. More seriously, the case split in the proof does not cover all vectors since the complement of “every coordinate of $|\hat{\mathbf{s}}-n\hat{\mathbf{m}}|$ is at most $n^p$” is “some coordinate exceeds $n^p$”. Because the estimates in the proof depend on $\max_j |\hat{\mathbf{s}}_j - n\hat{\mathbf{m}}_j|$, splitting into cases according to whether this maximum is at most $n^p$, between $n^p$ and linear growth of the form $\delta n$, or at least $\delta n$ repairs the argument.
Example 5.13 (Pages 234 – 235): The variables $z_{t+1}$ and $z_t$ are accidentally used for $\color{red}z_{d+1}$. Furthermore, the final component of the logarithmic gradient at $\sigma$ should be ${\color{red}-\rho h^{\prime}(\rho)}$, meaning $$\mathbf{m} = \left(-\frac{\rho}{h^{\prime}(\rho)},\dots,-\frac{\rho^d}{h^{\prime}(\rho)}, 1\right),$$ and in the displayed equation for the maximal coefficients $A_n$, the factor $(2\pi n)^{d/2}$ should be $(2\pi n)^{{\color{red}-d/2}}$.
Page 257: In the first display, the final right-hand side should be $(-1)^rS(1)^r{\color{red}2^r}(1+x_{r+1})\cdots(1+x_d)\,r!/(a_1\cdots a_r)$.
Examples 7.3, 7.8, 7.9, and 7.10 (Pages 276, 285, 287, and 289): The ideal $I$ at the top of page 285 lists the polynomial $xH_x - \lambda$ twice (which does not change the ideal, but is a typographical error); the same duplicated polynomial also appears in the ideals displayed in Example 7.9 and Example 7.10. Furthermore, in all four examples the polynomial testing minimality should be $\tilde{H}(t,w,x,y,z) = H(tw,tx,ty,{\color{red}tz})$.
Example 7.4 (Page 278): The factored form of $F$ should be $$ \frac{1}{1-x-y} \times \frac{1}{20 - x - {\color{red}40}y - \frac{1}{1-x-y}}. $$
Example 7.5 (Page 279): There is an extra negative sign, with the minimal critical points supposed to have coordinates $1/3\pm i/(3\sqrt{3})$. The final asymptotic result is correct as presented.
Definition 7.6 (Page 279): The Sturm sequence recursion $g_n = -\operatorname{rem}(g_{n-2}, g_{n-1})$ should start at $n = {\color{red}2}$.
Example 7.6 (Page 280): The exponential growth in the final formula should be $\left(\frac{1}{s^{2s}(1-s)^{{\color{red}2-2s}}}\right)^n$.
Definition 7.9 (Page 282): In the reverse graded lexicographic example, the final two monomials should be switched, to make $z_1^2z_2^3$ last. The first relation symbol is also misprinted as $\succeq_{rc}$.
Page 289: In the statement “it is clearly natural to encode the values of each $z_j/P^{\prime}(u)$ instead of just $z_j$” the division should be the multiplication $z_j\,{\color{red}\cdot}\,P^{\prime}(u)$.
Page 291: The final paragraph before Section 7.3.2.2 should say $\Phi_q$ divides the resultant of the polynomials ${\color{red}TP^{\prime}(u) - Q_q(u)}$ and $P(u)$.
Page 293: Below Lemma 7.5, the polynomial $B(T) = R(\sqrt{T})R(-\sqrt{T})$ should be $B(T) = {\color{red}R(T^2)}$.
Definition 7.16 (Page 296): The Euclidean norm needs to include the constant in its summation, $$\lVert P\rVert_2 = \left(\sum_{j={\color{red}0}}^D|c_j|^2\right)^{1/2}.$$
Equation (7.12) (Page 299): The final equality should read $$\sqrt{3}\,|\Delta(P)|^{{\color{red}1/2}} \leq |r_a - r_b|D^{(D+2)/2}M(P)^{D-1}.$$ The corrected chain still yields Proposition 7.7(ii) exactly as stated.
Problem 7.4 (Page 302): The rational function $1/(1-(1+z)(x+y+xy))$ should actually be ${\color{red}1/(1 - z(1+x)(1+y)(1+y+xy))}$ to represent the Apéry-$\zeta(2)$ numbers.
Page 314: The set $\mathcal{B}$ is the collection of all connected components of $\mathcal{M}_{\mathbb{R}}$, so the running example should say that $\{B_0, B_1, B_2\}$ is the set of bounded components but not that it equals $\mathcal{B}$.
Page 314: The statement $\ell_j(\mathbf{x}+i\mathbf{y}) = \ell_j(\mathbf{x}) + i\ell_j(\mathbf{y})$ is false since $\ell_j$ has the constant term. It should state $\ell_j(\mathbf{x}+i\mathbf{y}) = \ell_j(\mathbf{x}) {\color{red} \;-\; i(b^{(j)}\cdot \mathbf{y})}$.
Proof of Lemma 8.1 (Page 317): The sentence after Equation (8.9) should say $\lambda \in \mathbb{R}^{{\color{red}s}}$ and $v = {\color{red}(\lambda,0)M}$.
Remark 8.7 (Page 320): Since $\nabla_{\log}\ell_k(\sigma) = -\sigma\odot b^{(k)}$, the equivalence should say $\mathbf{r}$ lies in the positive real span of the ${\color{red}\text{negated}}$ logarithmic gradients $-\nabla_{\log}\ell_{k_j}$.
Equations (8.19) – (8.20) (Page 328): The fiber basepoints should be $\sigma_{12} - \epsilon v_\kappa = \mathbf{1} - \epsilon {\color{red}M^{-1}}\boldsymbol{\kappa}$ for $\boldsymbol{\kappa}\in\{\pm1\}^2$, as in the general construction on page 331. Equation (8.21), after the change of variables, is correct.
Page 329: In Equation (8.21) and below $\det M$ should be replaced by ${\color{red}|}\det M \, {\color{red}|}$.
Pages 332 – 334: The prefactor $\operatorname{sgn}(\boldsymbol{\sigma})$ from (8.26) is dropped in: Equation (8.29) on page 332, $I_{\boldsymbol{\sigma}}$ in Remark 8.10 on page 333, and the leading constant $C_0^{\boldsymbol{\sigma}}$ of Theorem 8.2 on page 334. This is harmless for singularities in the positive orthant (as in all the examples) but incorrect in general.
Equation (8.31) (Page 333): The sum should start at $j = {\color{red}0}$ (the residue has a constant term; with all $p_{k_j} = 1$ the printed sum starting at $j=1$ is empty, giving $P_\sigma \equiv 0$). Proposition 8.4 correctly sums from $j = 0$.
Proof of Lemma 8.4 (Page 334): The quantity $\sigma^2_{k+1}\cdots\sigma^2_d$ should be $\sigma^2_{\color{red}s+1}\cdots\sigma^2_d$.
Proof of Theorem 8.2 (Page 335): The argument should say that $\phi_\sigma(y)$ has strictly ${\color{red}\text{positive}}$ real part.
Example 8.6 (Page 340): The second asymptotic contribution should be $$[x^{pn}y^{qn}]\frac{1}{\ell_1\ell_3^2} \sim 12({\color{red}2q-p})n,$$ not $12(q-p)n$.
Page 342: The change of variables to get $\tilde\omega$ should be $(x,y) = (1,1) - {\color{red}M^{-1}}(p,q)$.
Page 344: The definition of $\tilde G$ should be $$\tilde G(\mathbf{z}) = \frac{G(\mathbf{z})}{z_1\cdots z_d\prod_{j>{\color{red}t}}\ell_j(\mathbf{z})^{p_j}}.$$
Page 347: The leading asymptotic term in Theorem 8.3 is half what it would be in the ${\color{red}\text{generic}}$ case.
Problem 8.4 (Page 348): The second endpoint of $\mathcal{A}_n = (-n^{-2/5}, n^{2/5})$ should be $n^{{\color{red}-}2/5}$.
Problem 8.9 (Page 349): The Cauchy–Binet display has the submatrix indices transposed: with $A$ an $m\times n$ matrix and $B$ an $n \times m$ matrix, it should read $$\det(AB) = \sum_{S\in S_m}\det(A_{[m],S})\det(B_{S,[m]}).$$
Page 360: The points $(i,i,-i)$ and $(-i,-i,i)$ in the stratum $\mathcal{S}_{1,2,3}$ should be $(i, {\color{red}1}, -i)$ and $(-i, {\color{red}1}, i)$.
Example 9.6 (Page 362): The boundary-return asymptotics should have $A_n = {\color{red}24\sqrt2}$ for even $n$.
Example 9.3 Continued (Page 362): This should be Example 9.4 Continued.
Example 9.4 Continued (Page 368): This should be Example 9.3 Continued. Also, $\boldsymbol{\varrho}$ is the unique minimizer of $|x^{{\color{red}-a}}y^{{\color{red}-b}}|$ on the boundary of the power series domain of convergence.
Page 379: The lacuna phenomenon holds in even dimensions at least four (not greater than four).
Example 10.2 (Page 390): The displayed asymptotics should be $$ c_n = \left(1+2^{d-1}\right)^n\, n^{(1-d)/2}\left(\frac{(2^{d-1}-1)\,{\color{red}(1+2^{d-1})^{(d-1)/2}}}{(2^d\pi)^{(d-1)/2}}\right)\left(1 + O\left(n^{-1}\right)\right). $$
Proof of Proposition 10.1 and Remark 10.2 (Page 394): The names of the positive and negative drift cases are swapped when describing which of $p_1$ and $p_2$ contribute to asymptotics.
Equation (10.6) (Page 399): The exponential term should be $\color{red}\bar{S}(\mathbf{p})^n$ instead of $S(\mathbf{p})^n$. The surrounding objects $\mathcal{H}_{\mathbf{p}}$, $\mathcal{E}$, and $\psi_{\mathbf{p}}$ all correctly use $\bar S$.
Proof of Theorem 10.2 (Page 401): If $Q(\hat{\mathbf{z}})$ is not identically zero then only ${\color{red}p_1} = (\mathbf{1}, \sqrt{B(1)/A(1)})$ is a minimal contributing singularity.
Problem 10.1 (Page 402): The problem should ask to enumerate walks ending on the hyperplanes $\{z_j = 0\}$ for $j \in V$ to match the given diagonal expression.
Problems 10.3, 10.4, 10.6, and 10.7 (Pages 404 – 405): In all four problems the displayed left-hand side $\sum_{n\geq0}c_{n+2}t^n$ should be $\sum_{n\geq0}c_nt^{{\color{red}n+2}}$. In addition, the main diagonal of the rational function displayed in Problem 10.7 counts the walks ending strictly above the $x$-axis.
The following are minor typos which can mostly be inferred from context.
Page 11: In the first sentence of the Asymptotics section, the Cauchy integral should be Equation (1.5).
Page 46: In the first sentence after Lemma 2.2, the Puiseux series should be roots of ${\color{red}P}(z,y)$.
Example 2.27 (Page 72): In the sentence just before the displayed equation for $f_n$ there is a $\Gamma(z)$ which should be $\Gamma_{\color{red}0}(z)$.
Page 95: Just before Lemma 3.1, the point $\mathbf{w}\in\mathbb{C}$ should be $\mathbf{w}\in\mathbb{C}^{\color{red}d}$.
Example 4.5 (Page 157): The asymptotics of $C(t)$ up to a constant factor were determined in Example 2.29 of Chapter 2 (not in Chapter 3).
Lemma 5.2 (Page 208): The quantities $\mathfrak{R}_j(\hat{\mathbf{z}})$ should be $\mathfrak{R}_{\color{red}k}(\hat{\mathbf{z}})$.
Proposition 5.2 (Page 209): In the final paragraph, the condition $H^{\mathfrak{s}}_{z_d}(\mathbf{w})\neq0$ should be $H^{\mathfrak{s}}_{z_d}(\mathbf{w}_{\color{red}j})\neq0$.
Page 215: In the first sentence after Remark 5.13, the reference to Theorem 5.28 should be to Theorem 5.3.
Example 5.5 (Page 217): The condition $a,b,c\in\mathbb{R}^d_{>0}$ should be $a,b,c>0$.
Proof of Proposition 5.12 (Page 238): The sets $U_i = \{\mathbf{y}\in\mathbb{P}^m : y_i = 1\}$ should be defined for $0\leq i\leq {\color{red}m}$.
Problem 5.4 (Page 242): Figure 5.4 is in Chapter 5, not Chapter 3 as written.
Page 252: The paragraph before Theorem 6.1 should reference $G(\mathbf{z}) = (1+z_1)\cdots(1+z_{\color{red}d})$.
Theorem 6.1 (Page 252): The quadrant $\mathbb{N}^2$ should be the orthant $\mathbb{N}^{\color{red}d}$.
Definition 7.2 (Page 264): The inequality defining $f=O(g)$ should hold for $\mathbf{n}\in\mathbb{N}^{\color{red}m}$.
Example 7.1 (Page 265): The Maple code should use := for assignment instead of = as written.
Page 270: The polynomial system defined by (7.3) – (7.7) contains the $4d+3$ variables $\mathbf{a}, \mathbf{b}, \mathbf{x}, {\color{red}\mathbf{y}}, \lambda_R, \lambda_I$, and $t$.
Lemma 8.4 (Page 334): The matrix $A$ is the $d\times(d-{\color{red}s})$ submatrix of $M^{-1}$ consisting of its $d-s$ rightmost columns.
Example 8.3 Continued (Page 347): The displayed integral $\int_{\mathbb{R}+i\epsilon}(e^{-48ny^2}/y)\,dt$ should end in $dy$.
Proof Sketch of Theorem 9.1 (Page 366): Both references to Section 8.3 of Chapter 8 should actually cite Section 8.2.
Page 368: The first log-gradient $(\nabla_{\log}H_2)(\sigma) = (-1,-2/3,-1)$ should be $(\nabla_{\log}H_{\color{red}1})(\sigma) = (-1,-2/3,-1)$.
Table 10.1 (Page 389): In the highly symmetric row, Theorem 6.1 is in Chapter 6 not Chapter 5.
Page 390: In the first sentence of the page, $b_k = \sum_{s\in\mathcal{S},\, i_k = 1} w_s$ should have the summation condition $\color{red}s_k = 1$.
Definition 10.2 (Page 397): In the first two displayed equations on page 397, the quantities $A_j’’$ and $B_j’’$ should be $A_{\color{red}k}’’$ and $B_{\color{red}k}’’$.
Page 401: In the displayed formula for $\mathcal{E}^2(P\tilde\psi)(\mathbf{0})$ the sum should be indexed by $\color{red}j$ instead of $i$.
Problem 10.4 (Page 404): The kernel method is described in Chapter 4 not Chapter 10.