Let $P_D$ be the vector space of polynomials in one variable with degree at most $D$. I say that a polynomial is degenerate if it has a degenerate critical point. In other words, I define the set $X$ of degenerate polynomials by $$ X_D = \{p\in P_D \,\mid\, \text{there exists } z\in\mathbb{C} \text{ such that } p'(z)=0\text{ and } p''(z) = 0\}. $$ I am curious about the nature of $X_D$.

Is $X_D\subset P_D$ contained in the zero set of some polynomial of the coefficients of $1,x,\dots,x^D$? If so, of what degree?

I am also interested in the generalization to the case of multivariate polynomials.

Example: Consider the case $D=2$. We associate to the polynomial $p(x) = c_2x^2 + c_1 x + c_0$ the coordinates $(c_0,c_1,c_2)$. The degenerate polynomials have the form $p(x) = a$ and $p(x)=a(x-b)^2$.

Thus we have the description $$ X_2 = \{(a,0,0) \mid a\in\mathbb{C}\} \cup \{(ab^2, -2ab, a)\mid a,b\in\mathbb{C}\}. $$ The first set describes a curve, and the second set describes an algebraic surface (which is cut out by the zero set of the polynomial $f(x,y,z) = y^2-4xz$). In this case the degree of this surface is $2$. The curve is actually a straight line, and is contained in the zero set of $f(x,y,z)=y$, for example. Thus we have $$ X_D \subset \{(c_0,c_1,c_2) \mid (c_1^2 - 4c_0c_2) \cdot c_2 = 0\}. $$

  • $\begingroup$ I don't understand what you're asking. Say $D=2.$ Then $X_D$ consists of the constant polynomials and the polynomials of the form $(x-c)^2.$ How does this relate to your question about zero sets? $\endgroup$ – saulspatz Mar 8 '18 at 4:28
  • $\begingroup$ Thanks for the clarification. I think the case $D=2$ makes for a good example, so I have edited the question. $\endgroup$ – felipeh Mar 8 '18 at 4:46

Let $q=p'$. Then $p$ has a critical point iff $q$ and $q'$ share a root. But $q$ and $q'$ share a root iff the discriminant of $q$ vanishes.

Since $p$ has degree $D$, $q$ has degree $D-1$. Consequently, the discriminant of $q$ is a homogeneous polynomial over $\mathbb{Z}$ of degree $2D-4$ in the coefficients of $q$.

Since the coefficients of $q$ are integer multiples of the coefficients of $p$, it follows that there is a polynomial over $\mathbb{Z}$ of degree $2D-4$ in the coefficients of $p$ such that $p$ has a critical point iff this polynomial vanishes.

  • $\begingroup$ Fantastic! Is there something similar for the multivariable case? $\endgroup$ – felipeh Mar 8 '18 at 5:07
  • $\begingroup$ Yes: the Macaulay resultant of the gradient and Hessian determinant of a polynomial of degree $d$ in $n$ variables. The resultant is a polynomial over $\mathbb{Z}$ of degree $n^2(d-1)^{n-1}(d-2) +n(d-1)^n$ (I think) in the coefficients of the original polynomial, and vanishes iff the original polynomial has a degenerate critical point (I think). $\endgroup$ – K B Dave Mar 8 '18 at 6:24

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