Geodesic
Completely reducible factors of harmonic polynomials of three variables
Matematičeskie trudy, Tome 24 (2021) no. 2, pp. 24-36.

Voir la notice de l'article provenant de la source Math-Net.Ru

We describe the divisors of complex valued homogeneous harmonic polynomials on $\mathbb R^{3}$ which are products of linear forms and characterize the homogeneous polynomials $p$ that admit a couple of linear forms $\ell_{1}$ and $\ell_{2}$ such that $\ell_{1}^{m}p$ and $\ell_{2}^{m}p$ are harmonic for some $m\in\mathbb N$. The latter gives an example of a pair of spherical harmonics whose set of common zeros has length that is compatible with the upper bound of this quantity for a single harmonic. 
@article{MT_2021_24_2_a1,
     author = {V. M. Gichev},
     title = {Completely reducible factors of harmonic polynomials of three variables},
     journal = {Matemati\v{c}eskie trudy},
     pages = {24--36},
     publisher = {mathdoc},
     volume = {24},
     number = {2},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2021_24_2_a1/}
}
TY  - JOUR
AU  - V. M. Gichev
TI  - Completely reducible factors of harmonic polynomials of three variables
JO  - Matematičeskie trudy
PY  - 2021
SP  - 24
EP  - 36
VL  - 24
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MT_2021_24_2_a1/
LA  - ru
ID  - MT_2021_24_2_a1
ER  - 
%0 Journal Article
%A V. M. Gichev
%T Completely reducible factors of harmonic polynomials of three variables
%J Matematičeskie trudy
%D 2021
%P 24-36
%V 24
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MT_2021_24_2_a1/
%G ru
%F MT_2021_24_2_a1
V. M. Gichev. Completely reducible factors of harmonic polynomials of three variables. Matematičeskie trudy, Tome 24 (2021) no. 2, pp. 24-36. http://geodesic.mathdoc.fr/item/MT_2021_24_2_a1/

[1] Burbaki N., Gruppy i algebry Li, Gl. IV-VI, Mir, M., 1972

[2] Burbaki N., Gruppy i algebry Li, kompaktnye veschestvennye gruppy Li, Mir, M., 1986

[3] Vinberg E. B., Onischik A. L., Seminar po gruppam Li i algebraicheskim gruppam, Nauka, M., 1988

[4] Gichev V. M., “Neskolko zamechanii o sfericheskikh garmonikakh”, Algebra i analiz, 20:4 (2008), 64–86

[5] Agranovsky M. L., “On a problem of injectivity for the Radon transform on a paraboloid”, Contemporary Math., 251, 1999, 1–14

[6] Agranovsky M. and Krasnov Y., “Quadratic divisors of harmonic polynomials in ${\mathbb R}^{n}$”, J. Anal. Math., 82 (2000), 379–395 | DOI | Zbl

[7] Bateman H., Erdélyi A., Higher Transcendental Functions, v. 2, McGraw-Hill Book Co, New York–London, 1953

[8] Courant R., Hilbert D., Methoden der Mathematischen Physik, v. 1, Verlag von Julius Springer, Berlin, 1931

[9] Gichev V. M., “Factorization of special harmonic polynomials of three variables”, Sib. Èlektron. Mat. Izv., 17 (2020), 1229–1312

[10] Logunov A., Malinnikova E., “On ratios of harmonic functions”, Adv. Math., 274 (2015), 241–262 | DOI | Zbl

[11] Logunov A., Malinnikova E., “Ratios of harmonic functions with the same zero set”, Geom. Funct. Anal., 26 (2016), 909–925 | DOI | Zbl

[12] Mangoubi D., Weller-Weiser A., “Harmonic functions vanishing on a cone”, Isr. J. Math., 230:2 (2019), 563–581 | DOI | Zbl

[13] Maxwell J. C., A Treatise on Electricity and Magnetism, v. 1, Dover Publ. Inc., New York, 1954 | Zbl

[14] Szegö G., Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., 23, Amer. Math. Soc., Providence, RI, 1959 | Zbl

[15] Sylvester J. J., “Note on spherical harmonics”, Phil. Mag., 2 (1876), 291–307 | DOI