A blog about math (mainly), computer tricks (sometimes) and jazz music.
In its classical statement, the theorem of Gelfand-Mazur is the following: Theorem 1 . — Let $K$ be a complete normed field which is an extension of the field $\mathbf C$ of complex numbers. Then $K=\mathbf C$. As in the history of any theorem, the reality is a bit more complicated. The theorem proved by Stanislas Mazur in 1938 concerns all fields $F$, extensions of the field $\mathbf R$ of real…
That formula of Cayley asserts that there are $n^{n-2}$ trees with vertices labeled $1,\dots,n$. This is a chapter in graph theory, in which graphs are “simple”. More precisely, let's call a graph structure on a set $V$ (called “vertices”) is the datum of a subset $E$ of 2-element subsets of~$V$, called “edges”. So an edge is of the form $\{a,b\}$, where $a$ and $b$ are distinct elements of $V$,…
This post is here to record a fact that ought to be better known and for which I ought to have known a proper proof beforehand. Let's start by defining two thirds of the terms of the title. By polynomial ring , I simply mean a ring of polynomials $A[T_1,T_2,\dots]$ over a commutative ring $A$, in any number of indeterminates. The most important case here will however be that of a ring of…
In the previous post , I had explained how I could prove a general version of the classic fact that transvections have determinant 1. Recall here that transvections in an $R$-module $M$ are linear maps of the form $x\mapsto x+f(x)v$, where $f$ is a linear form on $M$ and $v\in M$ is a vector such that $f(v)=0$. To be able to talk of determinant, I assumed that $M$ had a finite basis, so that…
A transvection in a $K$-vector space $V$ is a linear map $T(f,v)$ of the form $x\mapsto x + f(x) v$, where $f\in V^*$ is a linear form and $v\in V$ is a vector such that $f(v)=0$. It is known that such a linear map is invertible, with inverse given by $f$ and $-v$. More precisely, one has $T(f,0)=\mathrm{id}$ and $T(f,v+w)=T(f,v)\circ T(f,w)$. In finite dimension, these maps have determinant $1$…
Sometimes in mathematics, you are told about very elementary things of which you hadn't even thought. I was well aware of some “duality” between image and preimage, but I just learned from Anatole Dedecker (who learned it from Patrick Massot) about another “duality” between preimage and some other notion. Moreover, it appears that this new notion can be used for making slightly more natural a…
Let $R$ be a ring. Its Krull dimension is the supremum of the lengths $n$ of chains $P_0\subsetneq P_1 \subsetneq\dots\subsetneq P_n$ of prime ideals of $R$. When $R$ is a field, the null ideal is the only prime ideal, and it is a maximal ideal so that its Krull dimension is zero. When $R$ is a principal ideal domain which is not a field, there are two kinds of prime ideals: the null ideal is…
I've been formalizing mathematical theorems in Lean for some years now, and one of the major blocks is the difficulty of formalizing elementary results that mathematicians do not take the time to even state. For example, a mathematician's integer can implicitly be a natural number at a line and a real number at the next one, while proof assistants require that some “coercion” maps be introduced.…
Browsing through my Zotero database, I fall upon a paper by Harald Kuhn where he proposes an elementary proof of the Weierstrass approximation theorem. The proof is indeed neat, so here it is. Theorem. — Let $f\colon[0;1]\to\mathbf R$ be a continuous function and let $\varepsilon$ be a strictly positive real number. There exists a polynomial $P\in\mathbf R[T]$ such that…
This is an exposition of the proof of the inequality between arithmetic and geometric means given by A. Pełczyński (1992), “ Yet another proof of the inequality between the means ”, Annales Societatis Mathematicae Polonae. Seria II. Wiadomości Matematyczne , 29 , p. 223–224. The proof might look bizarre, but I can guess some relation with another paper of the author where he proves uniqueness of…
Recently, in the Zulip server for Lean users, somebody went with something that looked like homework, but managed to sting me a little bit. It was about irreducibility of polynomials with integer coefficients. Specifically, the guy wanted a proof that the polynomial $T^4-10 T^2+1$ is irreducible, claiming that the Eisenstein criterion was not good at it. What was to proven (this is what…
Kronecker's theorem of the title is the following. Theorem. — Let $\alpha\in\mathbf C$ be an algebraic integer all of whose conjugates have absolute value at most $1$. Then either $\alpha=0$, or $\alpha$ is a root of unity. This theorem has several elementary proofs. In this post, I explain the simple proof proposed by Gebhart Greiter in his American Mathematical Monthly note , adding details so…
Reading a book on philosophy of mathematics, even if it's written lightly, such as that one, Why is there philosophy of mathematics at all? by Ian Hacking, may have unexpected effects. The most visible one has been a poll that I submitted on Mastodon on December 4th. As you can read, there were four options: Numbers exist Unicorns exist Numbers have more existence than unicorns Neither numbers no…
The “combinatorial Nullstellensatz” is a relatively elementary statement due to Noga Alon (1999) whose name, while possibly frightening, really says what it is and what it is good for. (A freely available version is there .) Nullstellensatz is the classic name for a theorem of David Hilbert that relates loci in $F^n$ defined by polynomial equations and the ideals of the polynomial ring…
Divided powers are an algebraic gadget that emulate, in an arbitrary ring, the functions $x\mapsto x^n/n!$ for all integers $n$, together with the natural functional equations that they satisfy. One of them is a nice binomial theorem without binomial coefficients : denoting $x^n/n!$ by $x^{[n]}$, one has $$ (x+y)^{[n]}=\sum_{k=0}^n x^{[n-k]} y^{[k]}. $$ Another formula looks at what happens when…
$\gdef\abs#1{\lvert#1\rvert}$ There are many parts of number theory I don't know of, and today's story is about one I learnt very recently, at the intersection of number theory and computer science. It is about the natural numbers that we can recognize using a finite automaton. Finite automata and automatic functions Finite automata Finite automata are very elementary computers. These machines can…
Let $E$ and $F$ be normed vector spaces, over the real or complex numbers, and let $u\colon E\to F$ be a linear map. The continuity of $u$ is proved to be equivalent to the existence of a real number $c$ such that $\|u(x)\|\leq c \|x\|$ for every $x\in E$, and the least such real number is called the operator norm of $u$; we denote it by $\|u\|$. It defines a norm on the linear space $\mathscr…
Polynomials are an algebraic gadget, and one is rarely led to think about the topology a ring of polynomials should carry. That happened to me, though, more or less by accident, when María Inés de Frutos Fernández and I worked on implementing in Lean the evaluation of power series. So let's start with them. To simplify the discussion, I only consider the case of one inderminate. When there are…
Projective modules and flat modules are two important concepts in algebra, because they characterize those modules for which a general functorial construction (Hom module and tensor product, respectively) behave better than what is the case for general modules. This blog post came out of reading a confusion on a student's exam: projective modules are flat, but not all flat modules are projective.…
$\global\def\Card{\operatorname{Card}}\global\def\GL{\mathrm{GL}}\global\def\im{\operatorname{im}}\gdef\KVar{\mathrm{KVar}}$ Let $n$ be an integer and $F$ be a field. Nilpotent matrices $N\in \mathrm M_n(F)$ are those matrices for which there exists an integer $p$ with $N^p=0$. Their characteristic polynomial is $\chi_N(T)=T^n$, and they satisfy $N^n=0$, which shows that the set $\mathscr N_n$ of…
Let $T_1,\dots,T_n$ be indeterminates. For all $m\geq0$, denote by $S_m$ the $m$th elementary symmetric polynomial , given by \[ S_m = \sum_{i_1 \lt \dots \lt i_m} T_{i_1}\dots T_{i_m}. \] These are polynomials in $T_1,\dots, T_n$, and as their names suggest, these polynomials are symmetric, meaning that \[ S_m (T_{\sigma(1)},\dots,T_{\sigma(n)}) = S_m (T_1,\dots,T_n) \] for any permutation…
This is a swift adaptation of a thread on Mastodon written after having read the beginning of Mara Goyet's column in yesterday's edition of Le Monde (please don't spoil the game by jumping to it now). So look at this picture below. Does something strike you in it? The directions? The ball? The character? Nothing? 68 people voted, roughly one third found nothing, and two thirds saw that there's a…
I would like to tell here a story that runs over some 150 years of mathematics, around the following question: given a power series $\sum a_n T^n$ (in one variable), how can you tell it comes from a rational function? There are two possible motivations for such a question. One comes from complex function theory: you are given an analytic function and you wish to understand its nature — the…
$\gdef\ann{\mathop{\mathrm{ann}}} \gdef\Ass{\mathop{\mathrm{Ass}}}\gdef\Spec{\mathop{\mathrm{Spec}}}$ I would like to go back to a quite delicate question of commutative algebra, that of associated prime ideals of modules. In most textbooks (Bourbaki, Matsumura…), this concept is considered for modules over a noetherian ring, while it is also necessary to consider it in a greater generality for…
The following reflexion came out of my irrepressible need to understand why the 3 double transpositions in $\mathfrak S_4$, together with the identity, formed a group $V$. Of course, one might just say: “they are stable under multiplication, as one sees by computing the 4·3/2 = 6 different products”, but who makes this computation anyway? And since I wanted not only to understand this, but to…