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  "body": "\n## Statement\n\nLet $D = \\set{z \\in \\C : \\abs{z} \u003c 1}$ be the unit disc.\nLet $f$ be a [[meromorphic function|0ZZ7ZYDR8W]] on $D$ that is analytic at $0$, and let $P$ be its set of [[poles|066E7PJCYE]] in $D$.\nLet $[n/n]_f$ be the diagonal [[Padé approximants|1F16G4V64X]] of $f$ at $0$.\nThen there is an infinite sequence $n_1 \u003c n_2 \u003c \\cdots$ such that $[n_j/n_j]_f$ converges to $f$ [[uniformly|08VZ57PT26]] on every compact subset of $D \\setminus P$.\n\n## Remarks\n\nThe conjecture is false in this generality.\nIn 2003 D. S. Lubinsky published a counterexample built from the Rogers–Ramanujan continued fraction.\nConvergence of the full diagonal fails in general, because the approximants can have spurious poles that wander through the disc; this is why the conjecture only asks for a subsequence.\nA weaker form holds: for functions meromorphic in the whole plane, the diagonal sequence converges in logarithmic capacity (the Nuttall–Pommerenke theorem).\nFor columns of the [[Padé table|MDM8F48X18]] with a fixed number of poles, [[Montessus de Ballore's theorem|XQQWP9ZPEW]] gives uniform convergence.\n\n## Explanations\n\n### L1 human\n\nDiagonal Padé approximants, with equal degrees of numerator and denominator, are the most used rational approximations to a power series.\nThey often converge very well in practice, but sometimes an approximant has an unwanted pole in a bad place.\nThe conjecture predicted that one can always skip the bad approximants and keep infinitely many good ones that converge uniformly.\nThis turned out to be false, although convergence holds in a weaker averaged sense.\n",
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      "first_stated": "1961",
      "posed_by": "George A. Baker, John L. Gammel and J. G. Wills"
    },
    "kind": "conjecture",
    "msc": [
      "41A21",
      "30E10"
    ],
    "name": "Baker–Gammel–Wills conjecture",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Pad%C3%A9_approximant"
      }
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    "tag": "YPTNBRWRY7",
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      "1F16G4V64X",
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  "kind": "conjecture",
  "link": "https://tamnd.github.io/math2/node/YPTNBRWRY7/",
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  "markdown": "https://tamnd.github.io/math2/node/YPTNBRWRY7.md",
  "msc": [
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    "30E10"
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  "name": "Baker–Gammel–Wills conjecture",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/YPT/YPTNBRWRY7.md",
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  "tag": "YPTNBRWRY7",
  "uri": "m2://YPTNBRWRY7",
  "url": "https://tamnd.github.io/math2/wiki/baker-gammel-wills-conjecture/",
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    "tag": "YPTNBRWRY7",
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