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The family of quadratic functions $F_2(a,b,c)$, consisting of all functions of the form $f(x)=ax^2+bx+c$, has the nice property (call it P) that given any $f,g\in F_2$, there is a sequence of function transformations taking $f$ to $g$. (More informally, property P is just the property that the graphs of functions in $F_2$ all "look the same," where equivalence is modulo function transformations.) Proof: just complete the square. (More generally, the set of all parabolas in the plane has this property if the allowed transformations are extended to include not just function transformations, but rotations as well.) One reason property P is nice is that it reduces root-finding for functions in $F_2$ to a single normal form: once you can find the roots of the equation $x^2+c=0$, you can find the roots of any function in the family $F_2$.

Families of higher-degree polynomials, however, lack property P. That is one reason why root-finding for cubics is harder than root-finding for quadratics: not all cubic functions in $F_3(a,b,c,d)$ can be transformed via function transformations into the standard form $f(x)=x^3+d$ -- the graphs of general cubic functions can differ from the graph of $x^3$ "more seriously" than can be captured or explained by a function transformation. So what's the best case? It turns out (see Wikipedia's entry on reduction of the general cubic to a monic trinomial) that the graphs of cubic functions of the form $f(x)=x^3+px+q$ (those lacking quadratic terms) exhibit enough of the general structure to generate all other cubics via function transformations. Root-finding for cubics is then reduced, in some sense, to the "normal form" $f(x)=x^3+px+q$. And Cardano's trick gives a slick way of solving that normal form using the third-degree binomial expansion.

QUESTION: Is there a nice higher-order way to say what property P is? I wanted to say something like "the family F(a,b,c) is closed under function transformations," but that's definitely not right: every family of polynomial functions of fixed degree is closed under function transformations. Instead, Property P is analogous to something like path-connectedness: given any two functions $f,g\in F$, there is a "path" -- here, a sequence of function transformations -- from $f$ to $g$. Or better still: there is only "one orbit" of the family F under the "action" of function transformations.

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I would view this as "orbits" under a "group action". You'll know this is at least a true model if: (1) the transformations can be described independent of which function they are transforming (so "substitute x=x+1 to change f(x) to f(x+1)" and tschirnhaus transformations in general are ok, but "change this specific function to this specific function" is probably not ok), (2) you can un-apply the transformation. – Jack Schmidt Jun 26 '12 at 17:13
Thanks Jack -- I must have realized this at the exact time you commented (my latest edit includes this view in the last sentence). – symplectomorphic Jun 26 '12 at 17:16
If it was me choosing the group, I'd choose AGL(1,K) × AGL(1,K) where K is the field acting on the inside and outside of the function. In other words, I'd allow the transformations x' = ax + b, and y' = ay + b. If you want to allow rotations, you'll lose f(x) being a function, but the equations y = f(x) will transform to simpler equations, and the appropriate group is the called AGL(2,K), affine general linear group. This is sufficient to depress, rotate, and translate the functions, but less powerful than Tschirnhaus. All of these groups are called "substitution groups" in Burnside's books. – Jack Schmidt Jun 26 '12 at 17:23

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