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On the Topology of the Set of Nonnegative Forms That Are Not Sums of Squares
Abstract
We study the topology of the normalized space of real nonnegative forms of degree 2d in n variables that are not sums of squares. Whenever it is nonempty, we prove that it is path-connected but not acyclic, and in particular not contractible. We give explicit bounds on the degrees in which its reduced homology can be nonzero. In the two minimal nonempty cases, we show an embedding of an exterior algebra into the cohomology ring, and prove containment of the binary octahedral and binary icosahedral groups, respectively, in its fundamental group.
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BibTeX
@misc{illmer2026topology,
title = {{On the Topology of the Set of Nonnegative Forms That Are Not Sums of Squares}},
author = {Maximilian Illmer and Tim Netzer},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07252},
eprint = {2611.07252},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07252},
}
RIS
TY - UNPB
TI - On the Topology of the Set of Nonnegative Forms That Are Not Sums of Squares
AU - Maximilian Illmer
AU - Tim Netzer
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.07252
UR - https://arxiv.org/abs/2611.07252
AB - We study the topology of the normalized space of real nonnegative forms of degree 2d in n variables that are not sums of squares. Whenever it is nonempty, we prove that it is path-connected but not acyclic, and in particular not contractible. We give explicit bounds on the degrees in which its reduced homology can be nonzero. In the two minimal nonempty cases, we show an embedding of an exterior algebra into the cohomology ring, and prove containment of the binary octahedral and binary icosahedral groups, respectively, in its fundamental group.
ER -