When students write exams, they want to know their score as fast as possible. However, the system that is supposed to deliver this information to the students is usually slow, for a lot of reasons that I will not go into. Where I work, it has become quite usual to publish a pdf on the course website that lists the student ID together with the number of points they scored. Since student IDs are usually anonymous, most people don't see it as a problem. However, the more this practice is used, the less anonymous a student ID actually is: Once there are enough of these lists public, this data might theoretically be used to connect a student to his or her student ID only by knowing which courses he or she attended.
I propose a much better solution. I did not come up with this myself by the way, this is due to a brillian colleague of mine, but I still wanted to share it with the world. (more…)

*everywhere*on $R$, which has infinite size, but $f$ is not the zero polynomial. The statement is of course true if $R$ is a commutative integral domain.

`anc`

at the root of the submission package.
If you are a novice to uploading files to the arXiv, like me, this might be confusing. What is the *submission package*? I only ever submitted a single $\KaTeX$ file! Well, let me put it straight for you.
- In the directory with your `.tex`

file(s), make a directory called `anc`

.
- Place all your source code and stuff in that directory.
- Make a zip file containing all your LaTeX sources and the folder `anc`

.
- Upload that zip file to the arXiv.
Trust me - everything will be fine.
`ptest.c`

, but you will need to get nauty to perform the entire computerized proof.
*is*a blog after all, so I thought I'd let the 8 people who read it know that for the next six weeks, I will be attending a special semester on Algorithms and Complexity in Algebraic Geometry at the Simons Institute for the Theory of Computing in Berkeley. So. If you happen to be in the bay area, give me a shout.

**Claim.***For each $1\le k\le n_\lambda$, if the function $f:=f_{\lambda k}\in\mathbb{C}[X]$ is reducible, then there exist weights $\lambda_1,\ldots,\lambda_r\in\Lambda_X$ such that $\lambda$ is an $\mathbb N$-linear combination of the $\lambda_i$.* Indeed, about a year ago this statement was completely unclear to me. However, it's actually not that hard to see and I felt like sharing my proof. (more…)

**Theorem.**Let $X$ be a normal, separated, Noetherian scheme and $U\subseteq X$ a nonempty affine open subset. Then, $X\setminus U$ is pure of codimension one.

**Proof.**We need to show that for each generic point $y\in Y=X\setminus U$, the local ring $\mathcal{O}_{X,y}$ has dimension one. Let $y$ be the generic point of a component of $Y$. Denote by $\mathfrak{m}$ the maximal ideal of the local ring $A:=\mathcal{O}_{X,y}$. Let $S:=\mathrm{Spec}(A)$. The inclusion morphism $f:S\to X$ is affine because $X$ is separated (Lemma 1). Thus, $V:=f^{-1}(U)=S\setminus\{ \mathfrak{m} \}$ is an affine, open subscheme of $S$. The morphism on global sections $A=\mathcal{O}_S(S)\to\mathcal{O}_{V}(V)$, corresponding to the inclusion $V\to S$, is therefore injective. It can not be surjective because $V\ne S$. Pick some $a\in\mathcal{O}_{V}(V)\setminus A$. If we now had $\dim(A)>1$, then the prime ideals of height one $\mathfrak{p}\subseteq A$ would satisfy $\mathfrak{p}\ne\mathfrak{m}$, i.e. they would correspond to points contained in $V$. Consequently, we would have $a\in \mathcal{O}_{V,\mathfrak{p}} = A_{\mathfrak{p}}$. Because $A$ is a normal Noetherian domain, it is the intersection of all localizations at prime ideals of height one - this is Corollary 11.4 in Eisenbud's book. This yields the contradition that $a\in A$.

**Lemma 1.**If $f:X\to Y$ is a morphism of schemes with $Y$ separated and $X$ affine, then $f$ is an affine morphism.

**Proof.**This is exercise 3.3.6 in Qing Liu's book. Let $V\subseteq Y$ be an open affine subset. Proposition 3.9(f) in the same book tells you that there is a closed immersion $f^{-1}(V)\cong X\times_Y V\to X\times V$, so $f^{-1}(V)$ is isomorphic to a closed subscheme of an affine scheme, hence affine.