Alt's Problem — Draconitic Research

Abstract

We prove there are $1442$ four-bar coupler curves through nine generic points in the plane, and thus resolve Alt's problem. We obtain this proof in three steps. First, we identify the space of coupler curves with a Zariski open subset of Gr(3,6). Next, we formulate the polynomial system representing the nine-point path synthesis problem in these coordinates and modify it to obtain the mixed volume $5538$. Finally, we prove that $4096$ of the branches of the generic sparse polynomial system with that support escape the torus in the sparse limit. Thus, we obtain an upper bound of 5538−4096=1442 for the generic solution count. A lower bound of $1442$ is achieved via numerical certification on one instance.

Cite

BibTeX RIS

BibTeX

@misc{brysiewicz2026alts,
  title = {{Alt's Problem}},
  author = {Taylor Brysiewicz},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22003},
  eprint = {2609.22003},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22003},
}

RIS

TY  - UNPB
TI  - Alt's Problem
AU  - Taylor Brysiewicz
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.22003
UR  - https://arxiv.org/abs/2609.22003
AB  - We prove there are $1442$ four-bar coupler curves through nine generic points in the plane, and thus resolve Alt's problem. We obtain this proof in three steps. First, we identify the space of coupler curves with a Zariski open subset of Gr(3,6). Next, we formulate the polynomial system representing the nine-point path synthesis problem in these coordinates and modify it to obtain the mixed volume $5538$. Finally, we prove that $4096$ of the branches of the generic sparse polynomial system with that support escape the torus in the sparse limit. Thus, we obtain an upper bound of 5538−4096=1442 for the generic solution count. A lower bound of $1442$ is achieved via numerical certification on one instance.
ER  -