Tricki

Proving linear independence of polynomials asymptotically

Quick description

If you need to prove that a family of polynomials are linearly independent, examine their asymptotic behaviour as the variables tend to .

Prerequisites

High-school algebra

General discussion

 Contributions wanted This section could use additional contributions. This article needs more (and better) examples. Is there a traceable origin to this trick? Is it an example of a more general trick?

If you need to prove linear independence of a big family of frightening-looking polynomials over or over , replace your variables with polynomials in with real numbers, with for or some permutation thereof, and examine the leading terms of your polynomials as tends to . If each polynomial in your family behaves differently `at ', you know that they cannot be linearly dependent— and only the leading terms need to be considered! Experiment with different substitutions in order to uncover the relevant asymptotic properties.

Example 1

This example is from Section 4 of math/0511602.

Set

where . We would like to show that the following polynomials are linearly independent over

with and for some fixed .

We first make the substitution

where are real numbers and . Although other substitutions were also possible, we chose this substitution so as to keep expressions of the form (with ) as simple as possible, because the polynomials , , and which make up are given in terms of such expressions. Thus

.

A routine calculation gives the leading term of as . This implies that the only possible linear relations between the 's are between those having the same value of .

Repeat the same trick again, this time making the substitution

where this time are real numbers satisfying . Again, we chose this substitution to keep the differences as simple as possible, while being different from the first substitution. In this case

This time the leading term of in terms of these new variables is . This implies that the only possible linear relations between the 's are between those having the same value of . Combining both results proves linear independence of the 's.

The following comments were made inline in the article. You can click on 'view commented text' to see precisely where they were made.

Am I right in thinking that

Am I right in thinking that the words "are linearly independent" are missing at the end of this sentence?