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Lower degree by increasing dimension (or vice-versa)

Quick description

When dealing with a high degree expression in one variable it is possible, and usually advantageous, to convert it into a low degree expression in many variables. On the other hand, it may sometimes be useful to hide all but one variables of an expression in n variables by lowering dimension (at the implicit expense of degree).

Prerequisites

Basic algebra.

Example 1: ode

Consider the following differential equation of degree n in one variable: y^{(n)}(t)=f(y^{(n-1)}(t),\dots ,y'(t),y(t),t). One may turn it into a degree one system of equations in n variables simply by renaming =t, =y(t),\dots, =y^{(n-1)}(t), obtaining: (t_1)' = 1

(t_2)' = t_3

(t_3)' = t_4

\vdots

(t_{n-1})' = t_n

(t_n)' = f(t_{n-1},\dots ,t_1)

The same could be done as above with a polynomial f of degree n in one variable X, instead of this differential equation, obtaining a system of polynomials of degree one in n variables X_1,\dots ,X_n.

Example 2: field

Let K be a function field in n variables over a field F. So K/F(x_1,\dots ,x_n) is a finite algebraic extension. If one is interested only with K and not in F, then one can replace F with F(x_2,\dots ,x_n) and view the function field K/F in n variables as a function field K/F(x_2,\dots ,x_n) in one variable.

General discussion

The pattern is clear from these examples: renaming dependent quantities to turn them into extra variables leads to a lower degree. This allows to apply standard degree-one techniques to degree-n problems.