Detecting integral polyhedral functions
Confluentes Mathematici, Volume 1 (2009) no. 1, pp. 87-109.

We study the class of real-valued functions on convex subsets of ℝn which are computed by the maximum of finitely many affine functionals with integer slopes. We prove several results to the effect that this property of a function can be detected by sampling on small subsets of the domain. In so doing, we recover in a unified way some prior results of the first author (some joint with Liang Xiao). We also prove that a function on ℝ2 is a tropical polynomial if and only if its restriction to each translate of a generic tropical line is a tropical polynomial.

Published online:
DOI: 10.1142/S1793744209000031
Kiran S. Kedlaya 1; Philip Tynan 1

1
@article{CML_2009__1_1_87_0,
     author = {Kiran S. Kedlaya and Philip Tynan},
     title = {Detecting integral polyhedral functions},
     journal = {Confluentes Mathematici},
     pages = {87--109},
     publisher = {World Scientific Publishing Co Pte Ltd},
     volume = {1},
     number = {1},
     year = {2009},
     doi = {10.1142/S1793744209000031},
     language = {en},
     url = {https://cml.centre-mersenne.org/articles/10.1142/S1793744209000031/}
}
TY  - JOUR
AU  - Kiran S. Kedlaya
AU  - Philip Tynan
TI  - Detecting integral polyhedral functions
JO  - Confluentes Mathematici
PY  - 2009
SP  - 87
EP  - 109
VL  - 1
IS  - 1
PB  - World Scientific Publishing Co Pte Ltd
UR  - https://cml.centre-mersenne.org/articles/10.1142/S1793744209000031/
DO  - 10.1142/S1793744209000031
LA  - en
ID  - CML_2009__1_1_87_0
ER  - 
%0 Journal Article
%A Kiran S. Kedlaya
%A Philip Tynan
%T Detecting integral polyhedral functions
%J Confluentes Mathematici
%D 2009
%P 87-109
%V 1
%N 1
%I World Scientific Publishing Co Pte Ltd
%U https://cml.centre-mersenne.org/articles/10.1142/S1793744209000031/
%R 10.1142/S1793744209000031
%G en
%F CML_2009__1_1_87_0
Kiran S. Kedlaya; Philip Tynan. Detecting integral polyhedral functions. Confluentes Mathematici, Volume 1 (2009) no. 1, pp. 87-109. doi : 10.1142/S1793744209000031. https://cml.centre-mersenne.org/articles/10.1142/S1793744209000031/

[1] N. Bourbaki , Fonctions d’une Variable Réelle ( Hermann , 1958 ) .

[2] K. S. Kedlaya, Alg. Number Th. 1, 269 (2007).

[3] K. S. Kedlaya, Compos. Math. 145, 143 (2009), DOI: 10.1112/S0010437X08003783 .

[4] K. S. Kedlaya, p-adic differential equations (version of 19 Jan 2009), preprint at http://math.mit.edu/kedlaya/papers/ .

[5] K. S. Kedlaya, Good formal structures for flat meromorphic connections, I: Surfaces , arXiv:0811.0190v3 [arXiv] .

[6] K. S. Kedlaya and L. Xiao, Differential modules on p-adic polyannuli, arXiv:0804.1495v4, to appear in J. Institut Math. Jussieu .

[7] R. T. Rockafellar , Convex Analysis (Princeton Univ. Press , 1970) .

[8] D. F. Young, Trans. Amer. Math. Soc. 347, 1323 (1995), DOI: 10.2307/2154812 .

Cited by Sources: