This is the 2nd Volume of the CMS/CAIMS Books in Mathematics.

A preprint (almost the same as the published version) is available from arXiv.

**A supplement** – an “elementary proof” of the BKK bound.

## Errata

The following list is going to be updated as new misprints/errors are discovered. The page numbers refer to the published version. Please send me an email or leave a comment below if you discover any not on this list.

$\DeclareMathOperator{\scrA}{\mathcal{A}} \DeclareMathOperator{\kk}{\mathbb{K}} \DeclareMathOperator{\zz}{\mathbb{Z}}$

- Chapter VI, Section 1 (Page 149): This is embarrassing. I claimed that every morphism from $(\kk^*)^m \to (\kk^*)^n$ is a monomial map (with respect to any choices of coordinates on the source and the target) and consequently homomorphisms. This is of course false. The morphisms are indeed monomial maps,
*but the coefficients of the monomials can be different from $1.$*The correct statement is that every morphism from $(\kk^*)^m \to (\kk^*)^n$ is of the form

\[x \mapsto (c_1x^{\alpha_1}, \ldots, c_nx^{\alpha_n})\]

with $c_1, \ldots, c_n \in \kk^*.$ - Chapter VI, Section 7, Exercise VI.20 (Page 165): the condition that “there is $\beta_\alpha \in \zz^n$ such that … touches every
**edge**of $\mathcal{C}_\alpha$” should be changed to “there is $\beta_\alpha \in \zz^n$ such that … touches every**facet**of $\mathcal{C}_\alpha$” - Chapter VI, Section 11, Hint to Exercise VI.37 (Page 176): $\beta_0$ should be chosen in the
*relative interior*of $d\mathcal{P} \cap \mathbb{Z}^I$ - Chapter VII, Section 6, Proposition VII.30 (Page 194): The statement for assertion (2) is incorrect. The error is in the definition of $\pi_\nu$, which should be defined as $\phi_\nu^{-1} \circ \pi \circ \phi_\nu$ where $\phi_\nu$ is an isomorphism of $\zz^n$ which maps $\nu \mapsto (0, \ldots, 0, 1)$. That this definition is correct is clear from the proof.
- Chapter VIII, Section 2, The sentence preceding identity (74) in the proof of Proposition VIII.1 (Page 208): “$\dim(V(\tilde f))$ has dimension $n-1$” should be changed to “$V(\tilde f)$ has dimension $n-1$“
- Chapter IX, Section 3.2, immediately before Remark IX.7 and the statement of Theorem IX.8 (Page 223): “$\mathcal{M}’_0$” should be changed to “$\mathcal{M}_0(\mathcal{A}’)$“
- Chapter X, Section 2.1, several places in Example X.5 (Pages 244, 245):
- Appendix B (Pages 308, 309): Section 7 should come before Section 6.

## Omissions

I wish I had a little more time to polish the draft before the final submission. In particular, some more examples/explanation would have been nice, e.g.

- There should have been a preliminary discussion and picture of tangent spaces/singular points in Section III.13. That complex varieties are manifolds near nonsingular points should have been highlighted more.
- In Section III.13.4 there should have been an example of a parametrization at a nonsingular point of some non-rational curve.
- Example III.122 of Section III.15 should have been elaborated or may be an exercise should have been included regarding the fact that the degree of a (reduced) hypersurface on the projective space equals the degree of the defining polynomial.
- In Section IV.2.2 an example should have been given of a collection of ideals which is
*not*a sheaf. - There should have been examples to illustrate the point of the (one dimensional case of the) Nagata compactification in Section IV.2.4. In particular there should have been examples where extending the subscheme structure is not completely trivial.
- Section VII.4 should have included an example to illustrate that the subset of BKK non-degenerate polynomials is
*not*open, as opposed to the subset of $\scrA$-non-degenerate polynomials. - The applications of Bernstein’s theorem to convex geometry by Khovanskii, Kaveh-Khovanskii, and Teissier should have been mentioned in Chapter VII or XII.

Gergő PintérDear Pinaki Mondal!

I met your book on mathoverflow, referred here: https://mathoverflow.net/questions/361856/analytic-and-algebraic-definitions-of-intersection-multiplicity-of-two-complex-a

Nowdays I work with a group of physicist and we use singularity theoretical methods to investigate degeneracy points of quantum systems. That’s why I need intersection multiplicity in general versions. I asked some questions about it on mathoverflow:

https://mathoverflow.net/questions/428444/intersection-multiplicity-via-parametrization-in-general

https://mathoverflow.net/questions/419532/the-quotient-of-an-algebra-with-an-ideal-whose-generators-are-decomposed-as-the

I found the closest arguments in your book How many zeros, but unfortunately I haven’t found the answers yet. What do you know about these questions?

Another – related – topic is Example IV.2. in your book, pg. 62. I don’t understand the role of lambda in (ii) . Actually is the second ideal defined in terms of x_ij and lambda, hence in an n^2+1 dimensional space?

This example relates to my other question: https://mathoverflow.net/questions/422909/reduced-structure-of-the-set-of-hermitian-matrices-with-multiple-eigenvalues

Would you like to talk about these questions at some time?

Best,

Gergő Pintér

pinter.gergo@ttk.bme.hu