Category Archives: Algebra

Zeros of random polynomials

Posted on 21/11/2017 by lamwk

Given a polynomial , where the coefficients are random, what can we say about the distribution of the roots (on )? Of course, it would depend on what “random” means. Here, “random” means that the sequence is an i.i.d. sequence … Continue reading →

Posted in Algebra, Complex analysis, Potential theory, Probability | Leave a comment

Hopf fibration double covers circle bundle of sphere

Posted on 09/11/2017 by KKK

Two days ago, I gave a seminar talk on Chern‘s proof of the generalized Gauss-Bonnet theorem. Here I record the answer to a question asked by one of my colleague during the talk. Although not directly related to the proof … Continue reading →

Posted in Algebra, Differential geometry, Group theory | 3 Comments

The Cauchy-Schwarz inequality and the Lagrange identity

Posted on 20/05/2017 by KKK

The classical Lagrange identity is the following: This can be proven by expanding and separating the terms into the cross-terms part and the non cross-terms part. The Lagrange identity implies the Cauchy-Schwarz inequality in . And when , this can … Continue reading →

Posted in Algebra, Group theory, Inequalities, Linear Algebra | 1 Comment

27 lines on a smooth cubic surface

Posted on 10/09/2016 by HL

Here describes two different proofs of a general smooth cubic surface containing exactly 27 lines. One approach uses blow-ups and the other one uses the Grassmannian. If I have time I will elaborate on the discussion.  

Posted in Algebraic geometry | Leave a comment

Toric perspective 1

Posted on 23/05/2015 by HL

This series of posts will be about toric varieties. The author is not sure if there is a part 2, but he still calls this part 1. This post is about computing the dimensions and degrees of popular toric varieties. … Continue reading →

Posted in Algebraic geometry, Combinatorics | 4 Comments

A Fibonacci-like sequence

Posted on 05/03/2015 by KKK

What’s the pattern of the following sequence? Can you guess the next term? The answer is

Posted in Algebra, Combinatorics | Leave a comment

High degree polynomial with few real roots

Posted on 08/12/2014 by HL

There are univariate polyomials that the number of non-real roots is significantly larger than that of real roots. The simplest example is where is odd. It has one real root and non-real roots. In this post we show an example … Continue reading →

Posted in Algebra, Calculus, Miscellaneous | Leave a comment

Polynomial Optimization 3: Why do we need generalize?

Posted on 11/10/2014 by HL

It has been a while since the last post. Let us recall what we have done. We study the unconstrained polynomial optimization problem ()           where is a real polynomial. This problem is equivalent to () … Continue reading →

Posted in Algebra, Applied mathematics, Optimization | Leave a comment

Polynomial Optimization 2: SOS and SDP

Posted on 21/03/2014 by HL

In this article we shall describe something called Gram-matrix method which can decompose a polynomial into sum of squares. The notation means is a square symmetric positive semidefinite matrix.  Proposition 1.          Let , be a polynomial … Continue reading →

Posted in Algebra, Applied mathematics, Optimization | Leave a comment

Polynomial Optimization 1: Motivation

Posted on 29/11/2013 by HL

(Nov 28, 2013) A (constrained) polynomial minimization problem is in the form  ()          subject to where are polynomials in with real coefficients. If and , then the above problem becomes , an unconstrained polynomial minimization problem. … Continue reading →

Posted in Algebra, Applied mathematics, Optimization | Leave a comment

Sum of squares of polynomials

Posted on 08/05/2013 by HL

Today I went to a talk delivered by a postdoc in the CS department about bilinear complexity. He raised this elementary result: Theorem. Every polynomial in (that means, is a polynomial in with complex coefficients) can be expressed as a … Continue reading →

Posted in Algebra | 5 Comments