Geodesic
On the Action of the Complex Monge--Amp\`ere Operator on Piecewise Linear Functions
Funkcionalʹnyj analiz i ego priloženiâ, Tome 48 (2014) no. 1, pp. 19-29.

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

The action of the mixed complex Monge–Ampère operator $(h_1,\cdots,h_k)\mapsto dd^ch_1\wedge\cdots\wedge dd^ch_k$ on piecewise linear functions $h_i$ is considered. The language of Monge–Ampère operators is used to transfer results on mixed volumes and tropical varieties to a broader context, which arises under the passage from polynomials to exponential sums. In particular, it is proved that the value of the Monge–Ampère operator depends only on the product of the functions $h_i$.
Keywords: Newton polytope, mixed volume, Monge–Ampère operator, exponential sum, tropical variety.
Mots-clés : pseudovolume 
@article{FAA_2014_48_1_a1,
     author = {B. Ya. Kazarnovskii},
     title = {On the {Action} of the {Complex} {Monge--Amp\`ere} {Operator} on {Piecewise} {Linear} {Functions}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {19--29},
     publisher = {mathdoc},
     volume = {48},
     number = {1},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2014_48_1_a1/}
}
TY  - JOUR
AU  - B. Ya. Kazarnovskii
TI  - On the Action of the Complex Monge--Amp\`ere Operator on Piecewise Linear Functions
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2014
SP  - 19
EP  - 29
VL  - 48
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2014_48_1_a1/
LA  - ru
ID  - FAA_2014_48_1_a1
ER  - 
%0 Journal Article
%A B. Ya. Kazarnovskii
%T On the Action of the Complex Monge--Amp\`ere Operator on Piecewise Linear Functions
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2014
%P 19-29
%V 48
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2014_48_1_a1/
%G ru
%F FAA_2014_48_1_a1
B. Ya. Kazarnovskii. On the Action of the Complex Monge--Amp\`ere Operator on Piecewise Linear Functions. Funkcionalʹnyj analiz i ego priloženiâ, Tome 48 (2014) no. 1, pp. 19-29. http://geodesic.mathdoc.fr/item/FAA_2014_48_1_a1/

[1] A. Esterov, “Tropical varieties with polynomial weights and corner loci of piecewise polynomials”, Mosc. Math. J., 12:1 (2012), 55–76 ; arXiv: 1012.5800 | DOI | MR | Zbl

[2] B. Ya. Kazarnovskii, “$c$-veery i mnogogranniki Nyutona algebraicheskikh mnogoobrazii”, Izv. RAN, ser. matem., 67:3 (2003), 23–44 | DOI | MR | Zbl

[3] D. N. Bernshtein, “Chislo kornei sistemy uravnenii”, Funkts. anal. i ego pril., 9:3 (1975), 1–4 | MR

[4] M. Brion, “Piecewise polynomial functions, convex polytopes and enumerative geometry”, Parameter spaces (Warsaw, 1994), Banach Center Publ., 36, Polish Acad. Sci., Warsaw, 1996, 25–44 | DOI | MR | Zbl

[5] B. Ya. Kazarnovskii, “O nulyakh eksponentsialnykh summ”, Dokl. AN SSSR, 257:4 (1981), 804–808 | MR | Zbl

[6] B. Ya. Kazarnovskii, “Mnogogranniki Nyutona i korni sistem eksponentsialnykh summ”, Funkts. anal. i ego pril., 18:4 (1984), 40–49 | MR | Zbl

[7] S. Alesker, “Hard Lefschetz theorem for valuations, complex integral geometry, and unitarily invariant valuations”, J. Differential Geom., 63:1 (2003) | DOI | MR | Zbl

[8] G. G. Gusev, Euler characteristic of the bifurcation set for a polynomial of degree $2$ or $3$, arXiv: 1011.1390 | MR

[9] B. Ya. Kazarnovskii, “Multiplikativnaya teoriya peresechenii i kompleksnye tropicheskie mnogoobraziya”, Izv. RAN, ser. matem., 71:4 (2007), 19–68 | DOI | MR | Zbl

[10] E. Bedford, B. A. Taylor, “The Dirichlet problem for a complex Monge–Ampere equations”, Invent. Math., 37:1 (1976), 1–44 | DOI | MR | Zbl

[11] B. Kazarnovskii, On piecewise pluriharmonic functions, arXiv: 1206.3741