{
  "id": "2108.02757",
  "version": "v1",
  "published": "2021-08-05T17:43:42.000Z",
  "updated": "2021-08-05T17:43:42.000Z",
  "title": "Generalized splines on graphs with two labels and polynomial splines on cycles",
  "authors": [
    "Portia Anderson",
    "Jacob P. Matherne",
    "Julianna Tymoczko"
  ],
  "comment": "25 pages",
  "categories": [
    "math.CO",
    "cs.NA",
    "math.AC",
    "math.NA"
  ],
  "abstract": "A generalized spline on a graph $G$ with edges labeled by ideals in a ring $R$ consists of a vertex-labeling by elements of $R$ so that the labels on adjacent vertices $u, v$ differ by an element of the ideal associated to the edge $uv$. We study the $R$-module of generalized splines and produce minimum generating sets for several families of graphs and edge-labelings: $1)$ for all graphs when the edge-labelings consist of at most two finitely-generated ideals, and $2)$ for cycles when the edge-labelings consist of principal ideals generated by elements of the form $(ax+by)^2$ in the polynomial ring $\\mathbb{C}[x,y]$. We obtain the generators using a constructive algorithm that is suitable for computer implementation and give several applications, including contextualizing several results in classical (analytic) splines.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-05T17:43:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25",
      "05C78",
      "05E16",
      "41A15"
    ],
    "keywords": [
      "generalized spline",
      "polynomial splines",
      "edge-labelings consist",
      "produce minimum generating sets",
      "adjacent vertices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}