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A quadratic or cubic polynomial (in two variables) can have at most one strict local minimum. A quartic polynomial can have up to five strict local minima [1]. So, how many strict local minima can a quintic polynomial have?

For quadratic and cubic polynomials, the possibility of having two strict local minima can be ruled out by evaluating the polynomial along the line connecting the two hypothetical minima. This however does not apply to polynomials of a higher order.

Denote $d$ the degree of a polynomial and $m$ ($n$) its number of nondegenerate local maxima (minima). Bézout's theorem implies that $m+n+s\leq (d-1)^2$, where $s$ is the number of nondegenerate saddles. What is more, by Corollary 6.9 in [2], $$m+n \leq \tfrac 1 2 d^2 -d+1.$$

By varying $d$, we can observe the following:

  • Quadratic polynomial: With $m+n\leq 1$, there is at most one local extremum.
  • Cubic polynomial: For this, $m+n \leq 2$. However, as noted earlier, at most one of these two potential local extrema can be a local minimum.
  • Quartic polynomial: Here, $m+n\leq 5$, and it is possible for all five potential local extrema to be local minima, as demonstrated here: Can a real quartic polynomial in two variables have more than 4 isolated local minima?
  • Quintic polynomial: In this case, $m+n\leq 8$.

Can all of the 8 potential local extrema of a quintic polynomial be of the minimum type, similar to the quartic polynomial case?

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  • $\begingroup$ Corollary 6.9 of the paper you cite in the question says that $m+n\le\frac{1}{2}d^2-d+1$, so $m+n\le8$ in your case. $\endgroup$
    – Peter Mueller
    May 20 at 13:37
  • $\begingroup$ @PeterMueller Thanks a lot for pointing this out, I edited the question accordingly to include this tighter inequality. $\endgroup$
    – Pavel Kocourek
    May 20 at 19:11
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    $\begingroup$ I foresee an infinite sequence of questions, as we go to sextic polynomials, septic polynomials, octic polynomials, .... $\endgroup$
    – Gerry Myerson
    May 21 at 0:35

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The paper Critical points of real polynomials, subdivisions of Newton polyhedra and topology of real algebraic hypersurfaces by Shustin claims to decide which triples $(m,n,s)$ are possible in all degrees (under the assumption that there are no singularities at infinity). It uses a cascade of notations and constructions, and I don't see at a first glance how to distill a polynomial for a possible triple $(m, n, s)$.

In this degree $5$ case, if I understand Theorem 3.1.6 correctly, we actually would have $m+n\le 8$ and $\text{min}(m, n)\ge2$.

So $m\le6$, and according to that Theorem, $(m,n,s)=(6,2,8)$ should be a possible case.

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