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Semialgebraic Cm solutions to a univariate system of linear inequalities with Nash forcing
Abstract
Let (Aij(x))1 ≤ i ≤ N, 1 ≤ j ≤ M be a matrix of univariate semialgebraic functions. We study the solvability of the system of linear inequalities ∑j=1M Aij(x)Fj(x)≤ fi(x) for i = 1, …, N, where we assume each forcing fi is a Nash function. We show that the Cm solvability of the system can be characterized by finitely many linear ordinary differential inequalities in f with semialgebraic coefficients depending only on (Aij). When the system is solvable, the solution may be chosen semialgebraic, and the proof gives a constructive procedure in the semialgebraic sense. We also study the semimodule structure of admissible forcings, establishing finite generation and lack thereof in various settings.
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BibTeX
@misc{jiang2026semialgebraic,
title = {{Semialgebraic $C^m$ solutions to a univariate system of linear inequalities with Nash forcing}},
author = {Fushuai Jiang and Garving K. Luli},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22772},
eprint = {2610.22772},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22772},
}
RIS
TY - UNPB
TI - Semialgebraic C^m solutions to a univariate system of linear inequalities with Nash forcing
AU - Fushuai Jiang
AU - Garving K. Luli
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.22772
UR - https://arxiv.org/abs/2610.22772
AB - Let (A_ij(x))_1 ≤ i ≤ N, 1 ≤ j ≤ M be a matrix of univariate semialgebraic functions. We study the solvability of the system of linear inequalities ∑_j=1^M A_ij(x)F_j(x)≤ f_i(x) for i = 1, …, N, where we assume each forcing f_i is a Nash function. We show that the C^m solvability of the system can be characterized by finitely many linear ordinary differential inequalities in f with semialgebraic coefficients depending only on (A_ij). When the system is solvable, the solution may be chosen semialgebraic, and the proof gives a constructive procedure in the semialgebraic sense. We also study the semimodule structure of admissible forcings, establishing finite generation and lack thereof in various settings.
ER -