Lower degree by increasing dimension (or vice-versa)

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Lower degree by increasing dimension (or vice-versa)

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[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|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 $t_1:=t$, $t_2:=y(t)$,$\dots$, $t_n:=y^{(n-1)}(t)$, 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|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.

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Lower degree by increasing dimension (or vice-versa)

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