The study of forms appears to grow "exponentially harder" with their degree as the following example seems to indicate.
Let us work over the field of complex numbers since the following statements become even more intricate over other fields!
A homogeneous form $F(x_0,\dots,x_n)$ of degree $d$ is said to be smooth if the ideal generated by its partial derivatives contains all monomials of sufficiently large degree.
A parametric solution of a form is a collection of polynomials $\mathbf{u}(\mathbf{y})=(u_i(y_1,\dots,y_n))_{i=0}^n$ such that "most" solutions of the form are values of the form $\mathbf{u}(\mathbf{a})$ for a suitable set of complex numbers $(a_1,\dots,a_n)$. By "most" solutions we mean all solutions that do not satisfy some independent other fixed form $G(x_0,\dots,x_n)$. (For those who know the terminology, this means that the associated variety is unirational.)
It was known classically that linear and quadratic forms have parametric solutions.
It is probably only less than 150 years ago or so that it was shown that smooth cubic forms have parametric solutions in 4 or more variables.
It is even more recently shown that smooth forms of any degree $d$ have parametric solutions in sufficiently many variables, where the number of variables grows more than exponentially with $d$. (It is worth finding out the exact number of variables required for $d=4$.) More recent efforts have strengthened these results somewhat, but exponential growth remains a "feature".
We can also ask whether there is a parametric solution where most solutions have a unique associated $(a_1,\dots,a_n)$. (This corresponds to asking whether the variety is rational.) Such a solution is called a rational solution.
It was known classically that such rational solutions are possible for linear and quadratic forms.
Smooth cubic forms in 4 or more variables that do have rational solutions have probably been known for a long time. It was only about 150 years ago that it was shown that no smooth cubic form in 3 variables has a rational solution and every smooth cubic form in 4 variables has a rational solution. It was only in the 70's that it was shown that there are smooth cubic forms in 5 variables that do not have a rational solution. For 6 or more variables the locus of smooth cubic forms that have a rational solution is not fully understood.
The situation for quartic forms (or, more generally, degree $\geq 4$) is more dire. There is not a single example known of a smooth quartic form that has a rational solution. However, it has not been proven that there is no such form either!
Ref: The following exposition appears to cover quite a bit of what is stated above. Details are in the references to this reference! "Unirationality of Hypersurfaces" by Robert Mijatovic