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 variables by lowering dimension (at the implicit expense of degree).
Example 1: ode
Consider the following differential equation of degree in one variable: . One may turn it into a degree one system of equations in variables simply by renaming , ,, , obtaining:
The same could be done as above with a polynomial of degree in one variable , instead of this differential equation, obtaining a system of polynomials of degree one in variables .
Example 2: field
Let K be a function field in variables over a field F. So is a finite algebraic extension. If one is interested only with and not in , then one can replace with and view the function field in variables as a function field in one variable.
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- problems.