{
  "annotation_id": "dcc2a76a-3f8f-4f41-b4b9-130921b72bfd",
  "date_created": "2026-02-17T05:53:24.169000Z",
  "date_modified": "2026-02-17T05:53:24.169000Z",
  "file_hash": "2bfbff33821fc547b6bd69a145819e72e85b9daffbdd065ec9a466cab66cb37d",
  "private": false,
  "record": {
    "abstract": "We consider three realization problems about monic real univariate polynomials without vanishing coefficients. Such a polynomial $P:=\\sum_{j=0}^db_jx^j$ defines the sign pattern $\\sigma (P):=({\\rm sgn}(b_d)$, $\\ldots$, ${\\rm sgn}(b_0))$. The numbers $p_d$ and $n_d$ of positive and negative roots of $P$ (counted with multiplicity) satisfy the Descartes\u0027 rule of signs. Problem~1 asks for which couples $C$ of the form (sign pattern $\\sigma$, pair $(p_d,n_d)$ compatible with $\\sigma$ in the sense of Descartes\u0027 rule of signs), there exist polynomials $P$ defining these couples. Problem~2 asks for which $d$-tuples of pairs $T:=((p_d,n_d)$, $\\ldots$, $(p_1,n_1))$, there exist polynomials $P$ such that $P^{(d-j)}$ has $p_j$ positive and $n_j$ negative roots. A $d$-tuple $T$ determines the sign pattern $\\sigma (P)$, but the inverse is false. We show by an example that $6$ is the smallest value of $d$ for which there exist non-realizable tuples $T$ for which the corresponding couples $C$ are realizable. The third problem concerns polynomials with all roots real. We give a geometric interpretation of the three problems in the context of degree $4$ polynomials.",
    "arxiv_id": "2601.10529",
    "authors": [
      "Vladimir Petrov Kostov"
    ],
    "categories": [
      "math.CA"
    ],
    "license": "http://creativecommons.org/publicdomain/zero/1.0/",
    "title": "Three realization problems about univariate polynomials",
    "url": "https://arxiv.org/abs/2601.10529",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "2d03aca5-7a63-4a0a-8d9a-94885b11d18a",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}