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 1d comment common roots of bivariate polynomial equationsTo be completely explicit: since the triangle has area 1/2, your equations have at most 2*1/2 = 1 isolated solution. Note that it is possible that you may have no solution at all for some specific coefficients (e.g. x^2y^3 + xy^2 + 1 = 0 and x^2y^3 + 2xy^2 + 2 = 0 have no common root). But what BKK theorem says is that for almost all of the coefficients you will have exactly one root. May15 awarded ● Enlightened May15 accepted Fano plane drawings: embedding PG(2,2) into the real plane May15 awarded ● Nice Answer May14 answered Fano plane drawings: embedding PG(2,2) into the real plane May6 comment Reference request: Samuel’s multiplicity and degreeHi Oleg, the comment of Steven Landsburg on this question: mathoverflow.net/questions/129673/… implies that Samuel's book should have a reference to at least the first question of yours. May5 comment Interpretation of multiplicity of a point @Steven: Thanks! May4 asked Interpretation of multiplicity of a point Apr27 comment Contracting a curve of negative self-intersection on a surfaceHi Philip, this article (of mine) gives a necessary and sufficient criterion for algebraicity in a special case: arxiv.org/abs/1301.0126 PS: I myself am interested in your Question 1, and I don't know of any other reference other than Grauert's original article, which is in German and therefore I can't read :( Feb14 comment A question on counting non-leading monomialsIs $a_{I,m}$ by definition $\lim_{k\to \infty} \sigma_{I,m}(k)/kh_I(k)$? If not, what is it? Feb8 revised jacobian polynomialadded 413 characters in body Feb8 answered jacobian polynomial Jan24 comment Normality condition on graded algebraHi Isac, A look at Chapter 5 of people.reed.edu/~iswanson/book/SwansonHuneke.pdf might help. Jan24 revised Normality condition on graded algebraadded 1024 characters in body Jan23 answered Normality condition on graded algebra Dec31 comment Contracting rational curves on surfaces and getting something non-algebraic@Jason: OK, I admit it wasn't such a good choice of words :) Would you prefer if I change "something non-algebraic" to "non-algebraic surfaces"? Dec30 comment Contracting rational curves on surfaces and getting something non-algebraic@Angelo: Thanks! But I knew of these (perhaps should have mentioned them in the question) and they do not contain (and as far as I can see, do not shed any light on the construction of) any such examples. Dec30 asked Contracting rational curves on surfaces and getting something non-algebraic Dec12 comment $d$ points on a curve which are in the base locus of a pencil of planesHi Francesco, the projection from a generic $p \in L$ may not be birational! The correct answer seems to be that either $L$ is a component of $C$, or there are hyperplanes $H_1, \ldots, H_m$ containing $L$ and curves $C_j \subseteq H_j$ of degree $d_j$ such that $1 \leq m \leq n$, $d_1 + \cdots d_m = d$ and $C = C_1 \cup \cdots \cup C_m$. Dec9 comment What are some examples of colorful language in serious mathematics papers?Oops! I just voted up and destroyed the magic of 27! I am really sorry, now if I downvote it becomes 26!