Random squarefree polynomials and random permutations and slightly non-random permutations

Influenced by Granville’s “Anatomy of integers and permutations” (already a play, soon to be a graphic novel) I had always thought as follows:  a polynomial of degree n over a finite field F_q gives rise to a permutation in S_n, at least up to conjugacy; namely, the one induced by Frobenius acting on the roots.  So the distribution of the degrees of irreducible factors of a random polynomial should mimic the distribution of cycle lengths of a random permutation, on some kind of equidistribution grounds.

But it’s not quite right.  For instance, the probability that a permutation is an n-cycle is 1/n, on the nose.

But the probability that a random squarefree polynomial is irreducible is about (1/n)(1-1/q)^{-1}.

The probability that a random polynomial, with no assumption of squarefreeness, is irreducible, is again about 1/n, the “right answer.”  But a random polynomial which may have repeated factors doesn’t really have an action of Frobenius on the roots — or at least it’s the space of squarefree monics, not the space of all monics, that literally has an etale S_n-cover.

Similarly:  a random polynomial has an average of 1 linear factor, just as a random permutation has an average of 1 fixed point, but a random squarefree polynomial has slightly fewer linear factors on average, namely (1+1/q)^{-1}.

Curious!

 

The best breakfast sandwich in Madison is the Breakfast Ricardo at Cafe Cortadito

Brand new at 418 E. Wilson, sharing space with the Cardinal bar.  Madison has not had a really good Cuban sandwich the whole time I’ve lived here; this seems very likely to have changed, if the Breakfast Ricardo at Cortadito is any indication; it’s a lot like a cubano, except it’s on a round Cuban sweet roll instead of grilled bread, and there’s an egg on it.  It is worth a special trip.

Cortadito is still working out some kinks; my croquettes were cold in the middle, and the kitchen forgot the guava pastelito I ordered with my sandwich.  I was going to comment that the fried plantains were more like potato chips than the dark, sweet long-cut plantains I was expecting, but it turns out that’s just the difference between tostones and maduros, and both are totally reasonable interpretations of “fried plantains.”

But who cares about that because as I mentioned this is the best breakfast sandwich in the city.

 

7 Apr 2013: Twins 4, Orioles 3

I took CJ to this game, his first at OPACY.  Great day out, perfect weather, but a terrible game, which both teams seemed to be trying their best to lose.  Joe Mauer dropped a foul popup.  Alexi Casilla made the last out for the Orioles on a weak grounder — but the pitcher bobbled it, and Casilla probably would have made first on the play if he hadn’t jogged half-heartedly out of the batter’s box.  And of course there was Adam Jones, who pulled up on a fly that was his to catch and let it drop three feet in front of him for a two-run double — yes, it was ruled a double, in an act of generosity so extravagant that the official scorer could have legally taken it off his taxes.  A week later, Jones would drop a fly ball against the Yankees to allow three runs to score in a 5-2 New York victory.  Jones looks like a really good center fielder, but the defensive metrics hate him, and I have to say the defensive metrics have the better of it at the moment.

Jason Hammel, in theory our ace, looked a lot worse than his line suggests; behind in the count all day, never seeming to find much of a rhythm.

So the Orioles, on the strength of this game, didn’t look like a good ballclub — but for the season as a whole, they’re holding their own against the powers of the AL East, and one can’t ask for much more than that.

After the game, CJ and I walked around the Inner Harbor, which has not changed at all since I was a kid, and seems to be just as crowded as popular and kid-pleasing as it ever has been.  OK, one change:  the Power Plant, which used to be a metal-oriented rock club, is now a Barnes and Noble.  Sort of strange, since metal is more popular and than it was when I was a kid, and books less so.  But Baltimore marches to its own beat.

Final note to self:  remember that, even in Baltimore, a crabcake is not the kind of food that’s likely to be good at the ballpark.

Imagery

Dan Sharfstein wins Guggenheim

Congratulations to Dan Sharfstein, who is one of this year’s Guggenheim Fellows!  I have written before about my admiration for Dan’s book The Invisible Line, and this seems a good occasion to say again — if you’re at all interested in the long, complicated history of race in America, buy the book and read it.  His new book will be about Oliver Otis Howard and the Freedmen’s Bureau.  This is the kind of project that requires long, deep research and painstaking thought.  I don’t know if we can Kickstarter things like this, and I’m glad we have the Guggenheim Foundation to help make them possible.

 

E.O. Wilson does not think math is unnecessary

This piece by E.O. Wilson has been much shared and much griped about in my circles, but I think it’s a case of a provocative headline (“Great Scientist ≠ Good at Math:  discoveries emerge from ideas, not number-crunching”) prepended by the WSJ to an essay that says something much more modest and defensible.  I’d paraphrase Wilson like this.   Being good in math is like being a good writer.  Everyone agrees:

• You can do great science and be a terrible writer;
• Being better at writing is a worthwhile aspiration for any scientist.

The conjunction of these two statements in no way feels like a denigration of writing.  Nor is Wilson denigrating math.

I’ve said this before but it’s important so I’ll keep saying it — when you write an opinion piece for a publication, you don’t write the headline — the editors do, and they’ll put whatever loosely relevant headline will generate the most clicks.

Encouraging!

The introduction to the textbook States of Matter, by David L. Goodstein:

Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906, by his own hand.  Paul Ehrenfest, carrying on the work, died similarly in 1933.  Now it is our turn to study statistical mechanics.

 

The Lonely Passion of Joey Heatherton

Hey, somebody has posted an acoustic cover of “The Lonely Passion of Joey Heatherton,” the greatest song (against much competition) by 90s Boston indie-rock heroes Prickly:

 

 

You can hear Prickly’s original (in cassette quality, sorry) at my earlier Prickly post.

Hiring at and from Wisconsin

Happy to report that the UW-Madison math department has added two more terrific young faculty members, both joining us next fall:  Daniel Erman in commutative algebra and algebraic geometry (seen previously on the blog counting smooth members in semiample linear systems over finite fields) and Uri Andrews in model theory.

In other awesome news, my former Ph.D. student Derek Garton will join the department at Portland State (his master’s degree alma mater!) as a tenure-track assistant professor.

Mean beef stroganoff

You know, my mom is a distinguished scientist, and she, too, made a mean beef stroganoff when I was a kid.  Of course, it was “skid road stroganoff” from Peg Bracken’s classic I Hate To Cook Book, friend to working scientists of all genders with small kids and twenty minutes to get dinner on the table.  Wouldn’t it be great if that’s what Yvonne Brill made, too?  I truly love this dish and I make it for my own family every once in a while, but the sad truth is that only AB and I actually like it, and AB is not picky.  

Skid Road Stroganoff

8 ounces uncooked noodles (about 4 1/2 cups)
1 beef bouillon cube
1 garlic clove, minced
1/3 cup onion, chopped
2 tablespoons cooking oil
1 pound ground beef
2 tablespoons flour
2 teaspoons salt
1/2 teaspoon paprika
Two 3-ounce cans mushrooms
1 can condensed cream of chicken soup, undiluted
1 cup sour cream
Chopped parsley

Start cooking those noodles, first dropping a bouillon cube into the noodle water. Brown the garlic, onion and crumbled beef in the oil. Ad the flour, salt, paprika, mushrooms and tomato paste, stir, and let it cook five minutes while you light a cigarette and stare sullenly at the sink. Then add the soup and simmer it – in other words, cook on low flame under boiling point – 10 minutes. Now stir in the sour cream – keeping the heat low, so it won’t curdle – and let it all heat through. To serve it, pile the noodles on a platter, pile the stroganoff mix on top of the noodles, and sprinkle chopped parsley around with a lavish hand.