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The visualization below shows yet another of Mueller's polynomials. This one with an interesting minimum a the origin, located close to a saddle point (see also https://mathoverflow.net/questions/442736):

The visualization below shows yet another of Mueller's polynomials. This one with an interesting minimum at the origin, located close to a saddle point (see also https://mathoverflow.net/questions/442736):

The answer is at most 5 and a few polynomials with this property (5 minima) is featured in the answer to this cross-posting: https://mathoverflow.net/questions/442736. There are visualizations of a few of them further below.

Answer: The maximum number of strict local minima of a quartic polynomial is $N=5$.

A few examples of such polynomials are provided in the answers to this cross-posting: https://mathoverflow.net/questions/442736, and they are visualized at the end of this post. In what follows will be argued that $N$ can not be greater than 5.

The following visualization shows a polynomial written as a sum of squares with an interesting minimum at the origin (Mueller):

Note: This doesn't look like a minimum but it is! There are shallow saddle points on either side of the minimum.