πm solutions of semialgebraic equations on curves
Abstract
Consider a system of equations A(x)Β· F(x) = B(x) on a subset X of βn, where A(x) and B(x) are matrix- and vector-valued semialgebraic functions on X, and the unknown F(x) is a field of vector-valued m-jets. We assume there is a solution which is the field of Taylor polynomials of order m on X of a πm vector-valued function f on βn, and ask whether we can find a πm semialgebraic solution f. Our main result is a positive answer in the case dim X = 1. The methods are based on an article of Fefferman and Luli for X β β2, and a secondary goal is to show that their approach applies in a much simpler way in the case dim X =1, or in the case of the semialgebraic Whitney extension problem in β2. Our results hold more generally for functions definable in an o-minimal expansion of β.
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BibTeX
@misc{bierstone2026πm,
title = {{$\mathcal C^m$ solutions of semialgebraic equations on curves}},
author = {Edward Bierstone and Jean-Baptiste Campesato},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07210},
eprint = {2610.07210},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05782645v1},
}
RIS
TY - UNPB
TI - π^m solutions of semialgebraic equations on curves
AU - Edward Bierstone
AU - Jean-Baptiste Campesato
PY - 2026
DA - 2026/10/05
PB - HAL
DO - 10.48550/arxiv.2610.07210
UR - https://hal.science/hal-05782645v1
AB - Consider a system of equations A(x)Β· F(x) = B(x) on a subset X of β^n, where A(x) and B(x) are matrix- and vector-valued semialgebraic functions on X, and the unknown F(x) is a field of vector-valued m-jets. We assume there is a solution which is the field of Taylor polynomials of order m on X of a π^m vector-valued function f on β^n, and ask whether we can find a π^m semialgebraic solution f. Our main result is a positive answer in the case dim X = 1. The methods are based on an article of Fefferman and Luli for X β β^2, and a secondary goal is to show that their approach applies in a much simpler way in the case dim X =1, or in the case of the semialgebraic Whitney extension problem in β^2. Our results hold more generally for functions definable in an o-minimal expansion of β.
ER -