Vandermonde Submatrices and Arithmetic Progressions

[This post, which is based on an ongoing discussion with Alex Russell and Ravi Sundaram, contains some unpublished results.]

Currently, we are asking whether all submatrices of the order-p Vandermonde matrix over a finite extension of GF(2) are invertible where p is prime. The answer is “no” in general: there are examples of fields where the Vandermonde matrix has a singular submatrix.

We can ask an easier(?) question, though. What happens if we randomly sample a set of columns and look into submatrices formed by a subset of the sampled columns. With a touch of beautiful insight, Professor Russell has connected Szemeredi’s theorem on arithmetic progressions with this question.

Let AP_k denote an arithmetic progression of length $latek k$. Let [N] := \{1, 2, \cdots, N\} for N \in \mathbb{N}.

The Szemerédi theorem says, any “sufficiently dense” subset S \subset [N] contains infinitely many AP_k for all k \in \mathbb{N}. A finitary version says: Fix your favourite k \in \mathbb{N}, \delta \in [0, 1]. Then,  there exists a natural N := N_{k, \delta} such that if you look any subset S \subset [N] of size at least \delta N, you will find an AP_k. Yet another version says:

Szemerédi’s Theorem. The size of the largest subset S \subset [N] without an AP_k cannot be too large; in particular, it is o(N).

Recall that a function f(x) is o(g) if it grows too slow compared to g(x), so that \lim_{N\rightarrow \infty}{f(x)/g(x) = 0}.

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Finding a Primitive p-th Root of Unity in a Finite Field, with C++ Code

To this day, no method of finding a generator of Z_p^* is known to be more efficient than essentially trying 2, then 3, and so on. Who cares? Well, the difficulty of breaking a certain public key cryptosystem (due to El Gamal) depends on the difficulty of working with generators of Z_p^*.Keith Conrad

An nth root of unity in a finite field F is an element r \in F satisfying r^n=1, where n is an integer. If n is the smallest positive integer with this property, r is called a primitive nth root of unity. If r is a primitive nth root of unity, then all elements in the set \mu_n = \{1, r, r^2, \cdots, r^{n-1}\} are also roots of unity. Actually, the set \mu_n form a cyclic group of order n under multiplication, with generator r.

Problem: Suppose you are given a finite field F=GF(2^d) of degree d, and you are promised that there indeed exists a primitive pth root of unity r\in F for p prime. Find r, and in particular, produce a C++code that finds it.

In what follows, we talk about how to find such a root and provide my C++ code; the code uses the awesome NTL library.

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Spectral Sparsification by Spielman and Srivastava: How They Connected the Dots

In this post, I will discuss my own understanding of the spectral sparsification paper by Daniel Spielman and Nikhil Srivastava (STOC 2008). I will assume the following:

  1. The reader is a beginner, like me, and have already glanced through the Spielman-Srivastava paper (from now on, the SS paper).
  2. The reader has, like me, a basic understanding of spectral sparsification and associated concepts of matrix analysis. I will assume that she has read and understood the Section 2 (Preliminaries) of the SS paper.
  3. The reader holds a copy of the SS paper while reading my post.

First, I will mention the main theorems (actually, I will mention only what they “roughly” say).

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