Referee Response Thank you very much for your comments and suggestions. The typos have all been addressed, and I list my responses below to the other comments. It should be made clear that the definition for excess number of an ideal is a definition of the author... You can mention that the definition is inspired by Fulton's defintion. ***I have included the following text on page 2: At times it will be more convenient to work with the equivalence number $$\Ecirc:=d_{1}\cdots d_{n}-\Edot.$$ This definition is inspired by the notion of the \emph{equivalence of an ideal} in \cite{intersectionBook} {[}Chapter $6${]}. This number is the difference between the Bezout bound and the excess number in the cases we consider. Typos: Addressed and fixed Final comment: Nice paper, well written. ***Thank you for the kind words and taking the time to read the paper. ------------------------------------- Second report: I believe the problem tackled in this paper is important, because of its potential applications to enumerative geometry. In principle I would be happy to recommend acceptance of this paper, but there are a few aspects in which the paper should be revised before I can do so. I see some technical problems, which I will explain in detail below, and which do not seem serious---they probably just require a little more attention in one of the main definitions. A different objection is that the author could be more explicit in relating what he does with Fulton's classical approach to excess intersection. While Segre classes are mentioned briefly in the introduction, a reader who is not too familiar with Fulton's theory may not be able to place the author's results in the context of this theory. It should be mentioned, for example, that Theorem 1 can be proven easily using Fulton-MacPherson intersection theory, and in fact doing so generalizes the result immediately to _any ideal, monomial or otherwise, generated by a regular sequence_ (!). Surely the author is aware of this fact, and so should be the intended audience for the paper. This does not detract from the proof given in this paper, which is a nice illustration of completely different techniques. The problem I see is that a naive reader would gather that results such as Theorem 1 are open problems beyond the reach of standard tools, and this is simply not the case. The same applies to the examples given in the section on algorithms: I feel that it should be mentioned that these examples can be worked out by hand without difficulties using the tools explained in Fulton's book. Again, this fact does not diminish the value of the paper, since the author's techniques are very different, but a naive reader should not garner the impression that numerical/computer-aided techniques are needed for such examples. Even within the realm of computer-aided algorithms, there are other ways to compute excess intersection: brute force (i.e., saturation) can be performed by most packages such as Macaulay2; and Segre classes of arbitrary ideals can also be computed with such packages, yielding alternative algorithms. The author could mention these alternatives at the beginning of section 3. ***I have included the following lines in section 3: "As mentioned in the introduction there are other ways to compute excess numbers with Segre classes. In addition, one can use off-the-shelf computer algebra software like \texttt{Macaulay2} to compute excess numbers by saturating the ideal of a $\Bsys$ by $\fI$. Also, the examples we present here can also be worked out by hand using Fulton-MacPherson intersection theory." I will now explain my technical objection. It has to do with the author's notion of `I-generic forms'. …. **!** I have provided a more detailed definition. I replaced I-generic forms with B_I -system, and talk about a general B_I -system. The examples in the second section have been adjusted to match this definition. I am quite happy with Example 7 etc. This is beautiful polyhedral geometry, beautifully explained. ***Thanks! I will add a few more comments below on specific items in the paper. Some of these will just summarize the lengthier discussion given above. p.1, last line, and through the paper: "generic" : I recommend choosing a different term. ** See **!** p.2, "...and a combinatorial proof of the theorem below." Mention that a more general statement has a straightforward proof using standard Fulton-MacPherson intersection theory. This does not detract from the paper, but it should not be passed in silence. **I have included the following sentence in the manuscript: "This theorem can be proven easily using Fulton-MacPherson intersection theory, and in fact doing so generalizes the result to any ideal generated by a regular sequence. But in the proof we present, we will see how $\Edot$ and $\Ecirc$ relate to the volume of a subdivided simplex." p.9, middle "While this may not always be true in practice..." : I am not sure what the author means. Excess intersection for monomial ideals is reduced (in this paper and in e.g., Aluffi's recent work) to computations of (generalized) volumes of polytopes, and this can be performed "in practice" in individual cases of considerable size, far exceeding what the brute force approach can do. Maybe the author means that there are few closed formulas for entire families of ideals, and indeed Theorem 1 is an example of such a formula. I suggest rephrasing this sentence. *** I removed that sentence completely. p.10, beginning of section 3.2: "the lower bounds can become sharp". Maybe comment as to whether this is suspected to be common or rare. *** I wish I could say more about this, but I don't yet have a good intuition yet on this yet. Bottom of the page: "...is unable to use the fundamental theorem..." : this sentence sounds strange: homotopies are usually not able or unable to do anything. In any case, why is this so? *** I rephrased the sentence to be "While the $\hit$-homotopy is easy to set up, the fundamental theorem of parameter continuation of isolated roots \cite{numBook} {[}Theorem $7.1.6${]} cannot be applied." p.12, last line of the text: "may be used in any situation". Does this mean "every"? Also, the whole sentence sounds a bit strange: "ideals...can have their intersection numbers computed...". It is usually better to choose active verbal forms rather than passive ones, when possible. ** I changed the last sentence to "We believe that the the $\hup$-homotopy can compute excess numbers of many other ideals defined by sparse forms in many unknowns." ***Thank you for taking the time to read the paper.