# Errata for An Invitation to Analytic Combinatorics

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.

Please email any errors you notice to the author.

• Due to editor error and a breakdown in Springer’s publication process the original version of the text 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.

• In Section 1.1.2 of the Introduction, 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.

• Corollary 2.1: 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: 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.

• Definition 3.1: The coefficients $f_{\mathbf{i}}$ should be in $\mathbb{K}$ instead of $\mathbb{K}^d$.

• Proposition 3.6: 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 $\mathbf{w}$.

• Example 3.12: The final term in the polynomial $P$ should be negated. Corrected, the polynomial is $P = z − xy − (x + y + xy)z \, {\color{red} - xz^2}$.

• Definition 3.9: The inequalities in the final sentence should all be strict. In other words, 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 $|z_j| < |w_j|$ for each $1 \leq j \leq d$.

• Theorem 5.1: 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).

• Page 211: The sentence before Lemma 5.3 has a gramatical error, and should say First, we show criticality is equivalent to the gradient of $\phi$ vanishing (not vanishings).

• Page 215: The first sentence should be “This variation is common in the literature.

• Proposition 5.4: 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.

• Example 5.6: 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: There is an extra factor of $s$ in the denominator of the leading constant. The leading constant should read $(rn)!(sn)!/\sqrt{2 \pi ae^{-a}(be^{-b}+ae^{-a}-ab)}$

• Lemma 5.7: 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 $F(\mathbf{w})$ then the point $|\mathbf{w}|$ with positive coordinates also lies in $\mathcal{V}$ (and is thus also minimal).

• Example 7.5: There is an extra negative sign in the minimal critical points: the coordinates ${\color{red}-}1/3\pm i/(3\sqrt{3})$ should be $1/3\pm i/(3\sqrt{3})$. The final asymptotic result is correct as presented.

• Example 7.8: The ideal $I$ at the top of page 285 lists the polynomial $xH_x - \lambda$ twice (which does not change the definition of the ideal, but is a typographical error).