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A note on isoparametric polynomials

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We show that any homogeneous polynomial solution of the eiconal type equation \(|\nabla F(x)|^2=m^2|x|^{2m-2}\), \(m\ge 1\), is either a radially symmetric polynomial \(F(x)=\pm |x|^{m}\) (for even \(m\)’s) or it is a composition of a Chebychev polynomial and a Cartan–Münzner polynomial.

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Acknowledgments

The author thanks Professor Zizhou Tang for bringing his attention to the paper [22] of Qi Ming Wang and the anonymous referee for his/her useful comments.

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Correspondence to Vladimir G. Tkachev.

Appendix: Some examples of CM-polynomials

Appendix: Some examples of CM-polynomials

Recall that a CM-polynomial can have only degree \(m=1,2,3,4\) or 6. We shortly outline all so far known CM-polynomials below (the only remained unsettled case is in dimension \(n=32\) of degree \(m=4\)), see the recent lection notes [8] for a full story. CM-polynomial of degrees \(m=1,2,3\) are the linear forms \(F=\langle x, e\rangle \) (with \(e\in \mathbb {R}^{n}\) being a unit vector), quadratic forms (2) for some subspace \(V\subset \mathbb {R}^{n}\) of dimension \(2\le \dim V\le [n/2]\), and Cartan cubics (3), respectively.

According to the recent classification [4, 6, 7] any CM-polynomial of degree four in dimension \(n\ne 32\) is either one of the constructed in [1, 10, 15, 16] and given explicitly by

$$\begin{aligned} F(x)=|x|^4-2\sum _{i=0}^{s}(x^\top A_i x)^2, \end{aligned}$$

where \(\{A_{i}\}_{0\le i\le s}\) is a system of symmetric endomorphisms of \(\mathbb {R}^{n}\) satisfying \(A_i A_j+A_jA_i=2\delta _{ij}\cdot \mathbf {1}_{\mathbb {R}^{n}}\), or one of the following two quartic forms

$$\begin{aligned} F_d(x)=\frac{1}{2}({{\mathrm{\mathrm {tr}}}}(Z\bar{Z})^2-\frac{3}{8}({{\mathrm{\mathrm {tr}}}}Z\bar{Z})^2),\quad d=1,2, \end{aligned}$$

where the matrix

$$\begin{aligned} Z=\left( \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} 0 &{} z_1 &{} z_2 &{} z_3 &{} z_4 \\ -z_1 &{} 0 &{} z_5 &{} z_6 &{} z_7 \\ -z_2 &{} -z_5 &{} 0 &{} z_8 &{} z_9 \\ -z_3 &{} -z_6 &{} -z_8 &{} 0 &{} z_{10} \\ -z_4 &{} -z_7 &{} -z_9 &{} -z_{10} &{} 0 \\ \end{array} \right) \end{aligned}$$

has real entries \(z_k=x_k\) in the case \(d=1\) and the complex entries \(z_k=x_k+ix_{10+k}\) in the case \(d=2\), and \(1\le k\le 10\).

Finally, there are two (harmonic) CM-polynomials of degree \(m=6\), each one in dimensions \(n=8\) and \(n=14\), [1, 9, 12].

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Tkachev, V.G. A note on isoparametric polynomials. Anal.Math.Phys. 4, 237–245 (2014). https://doi.org/10.1007/s13324-014-0067-z

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