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Suppose $x=(x_1,\dots, x_d) \in \mathbb{R}^{d}$ for an arbitrary dimension $d$. Let $p(x_1,\dots, x_d)$ be a degree 4 polynomial and consider the quartic defined by $p(x_1, \dots, x_d)=0$.

Is it possible to generically parameterize this quartic with $m<d$ variables? With 'generically' I mean up to possibly a few special cases that would require an individual method, but can be named explicitly.

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  • $\begingroup$ What are you parametrizing by? Polynomials? $\endgroup$
    – Stanley Yao Xiao
    Dec 4, 2017 at 9:17
  • $\begingroup$ For my purposes it could be any function $f(y_1,\dots, y_m), y\in A\subseteq\mathbb{R}^m$ with $A$ explicitly known and it doesn't have to be a polynomial. $\endgroup$
    – madison54
    Dec 4, 2017 at 9:22
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    $\begingroup$ And what do you mean by "explicitly known"? You can certainly show that such functions exist using the implicit function theorem... but aside from that it is not likely possible to get more explicit parametrizing functions. $\endgroup$
    – Stanley Yao Xiao
    Dec 4, 2017 at 9:50
  • $\begingroup$ I would like to explicitly write down the parameterizing function $f$. Are there classes of degree 4 polynomials (i.e. conditions on the coefficients) for which that is possible? $\endgroup$
    – madison54
    Dec 4, 2017 at 10:20
  • $\begingroup$ of course if your quartic is identically 0 on a subspace, then you can linearly change the variables and get a smaller number of variables.... $\endgroup$
    – Dima Pasechnik
    Dec 4, 2017 at 11:56

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