{
  "id": "2409.06866",
  "version": "v1",
  "published": "2024-09-10T21:05:18.000Z",
  "updated": "2024-09-10T21:05:18.000Z",
  "title": "The number of solutions of a random system of polynomials over a finite field",
  "authors": [
    "Ritik Jain"
  ],
  "comment": "13 pages",
  "categories": [
    "math.PR",
    "math.CO",
    "math.NT"
  ],
  "abstract": "We study the probability distribution of the number of common zeros of a system of $m$ random $n$-variate polynomials over a finite commutative ring $R$. We compute the expected number of common zeros of a system of polynomials over $R$. Then, in the case that $R$ is a field, under a necessary-and-sufficient condition on the sample space, we show that the number of common zeros is binomially distributed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-10T21:05:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random system",
      "finite field",
      "common zeros",
      "variate polynomials",
      "probability distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}