{
  "index": "2005-B-5",
  "type": "ALG",
  "tag": [
    "ALG",
    "ANA"
  ],
  "difficulty": "",
  "question": "Let $P(x_1,\\dots,x_n)$ denote a polynomial with real coefficients in the\nvariables $x_1, \\dots, x_n$, and suppose that\n\\[\n\\left( \\frac{\\partial^2}{\\partial x_1^2} + \\cdots + \\frac{\\partial^2}{\\partial\nx_n^2}\\right) P(x_1, \\dots,x_n) = 0 \\quad \\mbox{(identically)}\n\\] % Equation labelled (a) (label to the left of the equation) in AMM version.\nand that\n\\[\nx_1^2 + \\cdots + x_n^2 \\mbox{ divides } P(x_1, \\dots, x_n).\n\\] % Equation labelled (b) (label to the left of the equation) in AMM version.\nShow that $P=0$ identically.",
  "solution": "\\textbf{First solution:}\nPut $Q = x_1^2 + \\cdots + x_n^2$. Since $Q$ is homogeneous, $P$ is divisible\nby $Q$ if and only if each of the homogeneous components of $P$ is divisible\nby $Q$. It is thus sufficient to solve the problem in case $P$ itself is\nhomogeneous, say of degree $d$.\n\nSuppose that we have a factorization $P = Q^m R$ for some $m>0$, where\n$R$ is homogeneous\nof degree $d$ and not divisible by $Q$;\nnote that the homogeneity implies that\n\\[\n\\sum_{i=1}^n x_i \\frac{\\partial R}{\\partial x_i} = dR.\n\\]\nWrite $\\nabla^2$ as shorthand for $\\frac{\\partial^2}{\\partial\nx_1^2} + \\cdots + \\frac{\\partial^2}{\\partial x_n^2}$; then\n\\begin{align*}\n0 &= \\nabla^2 P \\\\\n&= 2mn Q^{m-1}R + Q^m \\nabla^2 R + 2 \\sum_{i=1}^n 2mx_i Q^{m-1}\n\\frac{\\partial R}{\\partial\nx_i} \\\\\n&= Q^m \\nabla^2 R + (2mn + 4md) Q^{m-1} R.\n\\end{align*}\nSince $m>0$, this forces $R$ to be divisible by $Q$, contradiction.\n\n\\textbf{Second solution:}\n(by Noam Elkies)\nRetain notation as in the first solution.\nLet $P_d$ be the set of homogeneous\npolynomials of degree $d$, and let $H_d$ be the subset of $P_d$\nof polynomials killed by $\\nabla^2$, which has dimension\n$\\geq \\dim(P_d) - \\dim(P_{d-2})$; the given problem amounts to showing\nthat this inequality is actually an equality.\n\nConsider the operator $Q \\nabla^2$ (i.e., apply $\\nabla^2$ then multiply\nby $Q$) on $P_d$; its zero eigenspace is precisely $H_d$.\nBy the calculation from the first solution, if $R \\in P_d$, then\n\\[\n\\nabla^2 (QR) - Q \\nabla^2 R = (2n+4d)R.\n\\]\nConsequently, $Q^j H_{d-2j}$ is contained in the eigenspace of $Q \\nabla^2$\non $P_d$ of eigenvalue\n\\[\n(2n+4(d-2j)) + \\cdots + (2n+4(d-2)).\n\\]\nIn particular, the $Q^j H^{d-2j}$ lie in distinct eigenspaces, so are\nlinearly independent within $P_d$. But by dimension counting,\ntheir total dimension is at least that of $P_d$.\nHence they exhaust $P_d$, and the zero eigenspace cannot have dimension\ngreater than $\\dim(P_d) - \\dim(P_{d-2})$, as desired.\n\n\\textbf{Third solution:}\n(by Richard Stanley)\nWrite $x = (x_1, \\dots, x_n)$ and $\\nabla = (\\frac{\\partial}{\\partial x_1},\n\\dots, \\frac{\\partial}{\\partial x_n})$.\nSuppose that $P(x) = Q(x)(x_1^2 + \\cdots + x_n^2)$. Then\n\\[\nP(\\nabla)P(x) = Q(\\nabla)(\\nabla^2)P(x) =0.\n\\]\nOn the other hand,\nif $P(x) = \\sum_\\alpha c_\\alpha x^\\alpha$ (where $\\alpha = (\\alpha_1,\n\\dots, \\alpha_n)$ and $x^\\alpha = x_1^{\\alpha_1} \\cdots\nx_n^{\\alpha_n}$), then the constant term of $P(\\nabla)P(x)$\nis seen to be $\\sum_\\alpha c_\\alpha^2$. Hence $c_\\alpha = 0$ for all\n$\\alpha$.\n\n\\textbf{Remarks:}\nThe first two solutions apply directly over any field of characteristic zero.\n(The result fails in characteristic $p>0$ because we may take\n$P = (x_1^2 + \\cdots + x_n^2)^p = x_1^{2p} + \\cdots + x_n^{2p}$.)\nThe third solution can be extended to complex coefficients\nby replacing $P(\\nabla)$ by its complex conjugate, and again the result\nmay be deduced for any field of characteristic zero.\nStanley also suggests\nSection 5 of the arXiv e-print \\texttt{math.CO/0502363} for\nsome  algebraic background for this problem.",
  "vars": [
    "x",
    "x_1",
    "x_n",
    "x_i"
  ],
  "params": [
    "P",
    "Q",
    "R",
    "m",
    "n",
    "d",
    "i",
    "j",
    "P_d",
    "H_d",
    "c_\\\\alpha",
    "\\\\alpha"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "vectorx",
        "x_1": "firstvar",
        "x_n": "lastvar",
        "x_i": "indexvar",
        "P": "polyvar",
        "Q": "quadvar",
        "R": "restvar",
        "m": "powerct",
        "n": "dimension",
        "d": "degree",
        "i": "indexone",
        "j": "indextwo",
        "P_d": "homopoly",
        "H_d": "harmonic",
        "c_\\alpha": "coefmulti",
        "\\alpha": "multiid"
      },
      "question": "Let $polyvar(firstvar,\\dots,lastvar)$ denote a polynomial with real coefficients in the variables $firstvar, \\dots, lastvar$, and suppose that\n\\[\n\\left( \\frac{\\partial^2}{\\partial firstvar^2} + \\cdots + \\frac{\\partial^2}{\\partial lastvar^2}\\right) polyvar(firstvar, \\dots,lastvar) = 0 \\quad \\mbox{(identically)}\n\\]\nand that\n\\[\nfirstvar^2 + \\cdots + lastvar^2 \\mbox{ divides } polyvar(firstvar, \\dots, lastvar).\n\\]\nShow that $polyvar=0$ identically.",
      "solution": "\\textbf{First solution:}\nPut $quadvar = firstvar^2 + \\cdots + lastvar^2$. Since $quadvar$ is homogeneous, $polyvar$ is divisible\nby $quadvar$ if and only if each of the homogeneous components of $polyvar$ is divisible\nby $quadvar$. It is thus sufficient to solve the problem in case $polyvar$ itself is\nhomogeneous, say of degree $degree$.\n\nSuppose that we have a factorization $polyvar = quadvar^{powerct} restvar$ for some $powerct>0$, where\n$restvar$ is homogeneous\nof degree $degree$ and not divisible by $quadvar$;\nnote that the homogeneity implies that\n\\[\n\\sum_{indexone=1}^{dimension} indexvar \\frac{\\partial restvar}{\\partial indexvar} = degree\\,restvar.\n\\]\nWrite $\\nabla^2$ as shorthand for $\\frac{\\partial^2}{\\partial\nfirstvar^2} + \\cdots + \\frac{\\partial^2}{\\partial lastvar^2}$; then\n\\begin{align*}\n0 &= \\nabla^2\\, polyvar \\\\\n&= 2\\,powerct\\,dimension\\, quadvar^{powerct-1} restvar + quadvar^{powerct} \\nabla^2 restvar + 2 \\sum_{indexone=1}^{dimension} 2\\,powerct\\,indexvar\\, quadvar^{powerct-1}\n\\frac{\\partial restvar}{\\partial\nindexvar} \\\\\n&= quadvar^{powerct} \\nabla^2 restvar + (2\\,powerct\\,dimension + 4\\,powerct\\,degree) quadvar^{powerct-1} restvar.\n\\end{align*}\nSince $powerct>0$, this forces $restvar$ to be divisible by $quadvar$, contradiction.\n\n\\textbf{Second solution:}\n(by Noam Elkies)\nRetain notation as in the first solution.\nLet $homopoly$ be the set of homogeneous\npolynomials of degree $degree$, and let $harmonic$ be the subset of $homopoly$\nof polynomials killed by $\\nabla^2$, which has dimension\n$\\geq \\dim(homopoly) - \\dim(polyvar_{degree-2})$; the given problem amounts to showing\nthat this inequality is actually an equality.\n\nConsider the operator $quadvar \\nabla^2$ (i.e., apply $\\nabla^2$ then multiply\nby $quadvar$) on $homopoly$; its zero eigenspace is precisely $harmonic$.\nBy the calculation from the first solution, if $restvar \\in homopoly$, then\n\\[\n\\nabla^2 (quadvar\\,restvar) - quadvar \\nabla^2 restvar = (2\\,dimension+4\\,degree) restvar.\n\\]\nConsequently, $quadvar^{indextwo}\\, harmonic_{degree-2\\,indextwo}$ is contained in the eigenspace of $quadvar \\nabla^2$\non $homopoly$ of eigenvalue\n\\[\n(2\\,dimension+4(degree-2\\,indextwo)) + \\cdots + (2\\,dimension+4(degree-2)).\n\\]\nIn particular, the $quadvar^{indextwo} H^{degree-2\\,indextwo}$ lie in distinct eigenspaces, so are\nlinearly independent within $homopoly$. But by dimension counting,\ntheir total dimension is at least that of $homopoly$.\nHence they exhaust $homopoly$, and the zero eigenspace cannot have dimension\ngreater than $\\dim(homopoly) - \\dim(polyvar_{degree-2})$, as desired.\n\n\\textbf{Third solution:}\n(by Richard Stanley)\nWrite $vectorx = (firstvar, \\dots, lastvar)$ and $\\nabla = (\\frac{\\partial}{\\partial firstvar},\n\\dots, \\frac{\\partial}{\\partial lastvar})$.\nSuppose that $polyvar(vectorx) = quadvar(vectorx)(firstvar^2 + \\cdots + lastvar^2)$. Then\n\\[\npolyvar(\\nabla)\\,polyvar(vectorx) = quadvar(\\nabla)(\\nabla^2)\\,polyvar(vectorx) =0.\n\\]\nOn the other hand,\nif $polyvar(vectorx) = \\sum_{multiid} coefmulti\\, vectorx^{multiid}$ (where $multiid = (multiid_1,\n\\dots, multiid_{dimension})$ and $vectorx^{multiid} = firstvar^{multiid_1} \\cdots\nlastvar^{multiid_{dimension}}$), then the constant term of $polyvar(\\nabla)\\,polyvar(vectorx)$\nis seen to be $\\sum_{multiid} coefmulti^2$. Hence $coefmulti = 0$ for all\n$multiid$.\n\n\\textbf{Remarks:}\nThe first two solutions apply directly over any field of characteristic zero.\n(The result fails in characteristic $p>0$ because we may take\n$polyvar = (firstvar^2 + \\cdots + lastvar^2)^p = firstvar^{2p} + \\cdots + lastvar^{2p}$.)\nThe third solution can be extended to complex coefficients\nby replacing $polyvar(\\nabla)$ by its complex conjugate, and again the result\nmay be deduced for any field of characteristic zero.\nStanley also suggests\nSection 5 of the arXiv e-print \\texttt{math.CO/0502363} for\nsome  algebraic background for this problem."
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "marigold",
        "x_1": "beekeeper",
        "x_n": "alligator",
        "x_i": "jellyfish",
        "P": "eucalyptus",
        "Q": "hippogriff",
        "R": "nightshade",
        "m": "butterscotch",
        "n": "lemongrass",
        "d": "parchment",
        "i": "tortoise",
        "j": "screwdriver",
        "P_d": "watercress",
        "H_d": "blacksmith",
        "c_\\alpha": "blueberry",
        "\\alpha": "salamander"
      },
      "question": "Let $eucalyptus(beekeeper,\\dots,alligator)$ denote a polynomial with real coefficients in the variables $beekeeper, \\dots, alligator$, and suppose that\n\\[\n\\left( \\frac{\\partial^2}{\\partial beekeeper^2} + \\cdots + \\frac{\\partial^2}{\\partial\nalligator^2}\\right) eucalyptus(beekeeper, \\dots,alligator) = 0 \\quad \\mbox{(identically)}\n\\]\nand that\n\\[\nbeekeeper^2 + \\cdots + alligator^2 \\mbox{ divides } eucalyptus(beekeeper, \\dots, alligator).\n\\]\nShow that $eucalyptus=0$ identically.",
      "solution": "\\textbf{First solution:}\nPut $hippogriff = beekeeper^2 + \\cdots + alligator^2$. Since $hippogriff$ is homogeneous, $eucalyptus$ is divisible\nby $hippogriff$ if and only if each of the homogeneous components of $eucalyptus$ is divisible\nby $hippogriff$. It is thus sufficient to solve the problem in case $eucalyptus$ itself is\nhomogeneous, say of degree $parchment$.\n\nSuppose that we have a factorization $eucalyptus = hippogriff^{butterscotch} nightshade$ for some $butterscotch>0$, where\nnightshade is homogeneous of degree $parchment$ and not divisible by $hippogriff$;\nnote that the homogeneity implies that\n\\[\n\\sum_{tortoise=1}^{lemongrass} jellyfish \\frac{\\partial nightshade}{\\partial jellyfish} = parchment nightshade.\n\\]\nWrite $\\nabla^2$ as shorthand for $\\frac{\\partial^2}{\\partial beekeeper^2} + \\cdots + \\frac{\\partial^2}{\\partial alligator^2}$; then\n\\begin{align*}\n0 &= \\nabla^2 eucalyptus \\\\\n&= 2 butterscotch lemongrass hippogriff^{butterscotch-1} nightshade + hippogriff^{butterscotch} \\nabla^2 nightshade + 2 \\sum_{tortoise=1}^{lemongrass} 2 butterscotch jellyfish hippogriff^{butterscotch-1} \\frac{\\partial nightshade}{\\partial jellyfish} \\\\\n&= hippogriff^{butterscotch} \\nabla^2 nightshade + (2 butterscotch lemongrass + 4 butterscotch parchment) hippogriff^{butterscotch-1} nightshade.\n\\end{align*}\nSince $butterscotch>0$, this forces $nightshade$ to be divisible by $hippogriff$, contradiction.\n\n\\textbf{Second solution:}\n(by Noam Elkies)\nRetain notation as in the first solution.\nLet $watercress$ be the set of homogeneous polynomials of degree $parchment$, and let $blacksmith$ be the subset of $watercress$ of polynomials killed by $\\nabla^2$, which has dimension $\\geq \\dim(watercress) - \\dim(watercress_{parchment-2})$; the given problem amounts to showing that this inequality is actually an equality.\n\nConsider the operator $hippogriff \\nabla^2$ (i.e., apply $\\nabla^2$ then multiply by $hippogriff$) on $watercress$; its zero eigenspace is precisely $blacksmith$.\nBy the calculation from the first solution, if $nightshade \\in watercress$, then\n\\[\n\\nabla^2 (hippogriff nightshade) - hippogriff \\nabla^2 nightshade = (2 lemongrass + 4 parchment) nightshade.\n\\]\nConsequently, $hippogriff^{screwdriver} blacksmith_{parchment-2screwdriver}$ is contained in the eigenspace of $hippogriff \\nabla^2$ on $watercress$ of eigenvalue\n\\[\n(2 lemongrass + 4 (parchment-2 screwdriver)) + \\cdots + (2 lemongrass + 4 (parchment-2)).\n\\]\nIn particular, the $hippogriff^{screwdriver} blacksmith^{parchment-2screwdriver}$ lie in distinct eigenspaces, so are linearly independent within $watercress$. But by dimension counting, their total dimension is at least that of $watercress$. Hence they exhaust $watercress$, and the zero eigenspace cannot have dimension greater than $\\dim(watercress) - \\dim(watercress_{parchment-2})$, as desired.\n\n\\textbf{Third solution:}\n(by Richard Stanley)\nWrite $marigold = (beekeeper, \\dots, alligator)$ and $\\nabla = (\\frac{\\partial}{\\partial beekeeper}, \\dots, \\frac{\\partial}{\\partial alligator})$.\nSuppose that $eucalyptus(marigold) = hippogriff(marigold)(beekeeper^2 + \\cdots + alligator^2)$. Then\n\\[\neucalyptus(\\nabla)eucalyptus(marigold) = hippogriff(\\nabla)(\\nabla^2)eucalyptus(marigold) =0.\n\\]\nOn the other hand, if $eucalyptus(marigold) = \\sum_{salamander} blueberry marigold^{salamander}$ (where $salamander = (salamander_1, \\dots, salamander_{lemongrass})$ and $marigold^{salamander} = beekeeper^{salamander_1} \\cdots alligator^{salamander_{lemongrass}}$), then the constant term of $eucalyptus(\\nabla)eucalyptus(marigold)$ is seen to be $\\sum_{salamander} blueberry^2$. Hence $blueberry = 0$ for all $salamander$.\n\n\\textbf{Remarks:}\nThe first two solutions apply directly over any field of characteristic zero. (The result fails in characteristic $p>0$ because we may take eucalyptus = (beekeeper^2 + \\cdots + alligator^2)^p = beekeeper^{2p} + \\cdots + alligator^{2p}.) The third solution can be extended to complex coefficients by replacing $eucalyptus(\\nabla)$ by its complex conjugate, and again the result may be deduced for any field of characteristic zero. Stanley also suggests Section 5 of the arXiv e-print \\texttt{math.CO/0502363} for some  algebraic background for this problem."
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "constant",
        "x_1": "fixedone",
        "x_n": "fixedlast",
        "x_i": "fixedindex",
        "P": "nonpolyn",
        "Q": "nonquadra",
        "R": "heteropoly",
        "m": "zeroexpnt",
        "n": "singular",
        "d": "degreeless",
        "i": "complete",
        "j": "inclusive",
        "P_d": "nonpolydeg",
        "H_d": "nonharmon",
        "c_\\\\alpha": "coeffbeta",
        "\\\\alpha": "betaindex"
      },
      "question": "Let $nonpolyn(fixedone,\\dots,fixedlast)$ denote a polynomial with real coefficients in the\nvariables $fixedone, \\dots, fixedlast$, and suppose that\n\\[\n\\left( \\frac{\\partial^2}{\\partial fixedone^2} + \\cdots + \\frac{\\partial^2}{\\partial\nfixedlast^2}\\right) nonpolyn(fixedone, \\dots,fixedlast) = 0 \\quad \\mbox{(identically)}\n\\]\nand that\n\\[\nfixedone^2 + \\cdots + fixedlast^2 \\mbox{ divides } nonpolyn(fixedone, \\dots, fixedlast).\n\\]\nShow that $nonpolyn=0$ identically.",
      "solution": "\\textbf{First solution:}\nPut $nonquadra = fixedone^2 + \\cdots + fixedlast^2$. Since $nonquadra$ is homogeneous, $nonpolyn$ is divisible\nby $nonquadra$ if and only if each of the homogeneous components of $nonpolyn$ is divisible\nby $nonquadra$. It is thus sufficient to solve the problem in case $nonpolyn$ itself is\nhomogeneous, say of degree $degreeless$.\n\nSuppose that we have a factorization $nonpolyn = nonquadra^{zeroexpnt} heteropoly$ for some $zeroexpnt>0$, where\nheteropoly is homogeneous\nof degree $degreeless$ and not divisible by $nonquadra$;\nnote that the homogeneity implies that\n\\[\n\\sum_{complete=1}^{singular} fixedindex \\frac{\\partial heteropoly}{\\partial fixedindex} = degreeless\\,heteropoly.\n\\]\nWrite $\\nabla^2$ as shorthand for $\\frac{\\partial^2}{\\partial\nfixedone^2} + \\cdots + \\frac{\\partial^2}{\\partial fixedlast^2}$; then\n\\begin{align*}\n0 &= \\nabla^2 nonpolyn \\\\\n&= 2zeroexpntsingular\\, nonquadra^{zeroexpnt-1}heteropoly + nonquadra^{zeroexpnt} \\nabla^2 heteropoly + 2 \\sum_{complete=1}^{singular} 2zeroexpnt fixedindex\\, nonquadra^{zeroexpnt-1}\n\\frac{\\partial heteropoly}{\\partial\nfixedindex} \\\\\n&= nonquadra^{zeroexpnt} \\nabla^2 heteropoly + (2zeroexpntsingular + 4zeroexpntdegreeless)\\, nonquadra^{zeroexpnt-1} heteropoly.\n\\end{align*}\nSince $zeroexpnt>0$, this forces heteropoly to be divisible by $nonquadra$, contradiction.\n\n\\textbf{Second solution:}\n(by Noam Elkies)\nRetain notation as in the first solution.\nLet $nonpolydeg$ be the set of homogeneous\npolynomials of degree $degreeless$, and let $nonharmon$ be the subset of $nonpolydeg$\nof polynomials killed by $\\nabla^2$, which has dimension\n$\\geq \\dim(nonpolydeg) - \\dim(nonpolyn_{degreeless-2})$; the given problem amounts to showing\nthat this inequality is actually an equality.\n\nConsider the operator $nonquadra \\nabla^2$ (i.e., apply $\\nabla^2$ then multiply\nby $nonquadra$) on $nonpolydeg$; its zero eigenspace is precisely $nonharmon$.\nBy the calculation from the first solution, if $heteropoly \\in nonpolydeg$, then\n\\[\n\\nabla^2 (nonquadra heteropoly) - nonquadra \\nabla^2 heteropoly = (2singular+4degreeless)heteropoly.\n\\]\nConsequently, $nonquadra^{inclusive} H_{degreeless-2inclusive}$ is contained in the eigenspace of $nonquadra \\nabla^2$\non $nonpolydeg$ of eigenvalue\n\\[\n(2singular+4(degreeless-2inclusive)) + \\cdots + (2singular+4(degreeless-2)).\n\\]\nIn particular, the $nonquadra^{inclusive} H^{degreeless-2inclusive}$ lie in distinct eigenspaces, so are\nlinearly independent within $nonpolydeg$. But by dimension counting,\ntheir total dimension is at least that of $nonpolydeg$.\nHence they exhaust $nonpolydeg$, and the zero eigenspace cannot have dimension\ngreater than $\\dim(nonpolydeg) - \\dim(nonpolyn_{degreeless-2})$, as desired.\n\n\\textbf{Third solution:}\n(by Richard Stanley)\nWrite $constant = (fixedone, \\dots, fixedlast)$ and $\\nabla = (\\frac{\\partial}{\\partial fixedone},\n\\dots, \\frac{\\partial}{\\partial fixedlast})$.\nSuppose that $nonpolyn(constant) = nonquadra(constant)(fixedone^2 + \\cdots + fixedlast^2)$. Then\n\\[\nnonpolyn(\\nabla)nonpolyn(constant) = nonquadra(\\nabla)(\\nabla^2)nonpolyn(constant) =0.\n\\]\nOn the other hand,\nif $nonpolyn(constant) = \\sum_{betaindex} coeffbeta constant^{betaindex}$ (where $betaindex = (betaindex_1,\n\\dots, betaindex_{singular})$ and $constant^{betaindex} = fixedone^{betaindex_1} \\cdots\nfixedlast^{betaindex_{singular}}$), then the constant term of $nonpolyn(\\nabla)nonpolyn(constant)$\nis seen to be $\\sum_{betaindex} coeffbeta^2$. Hence coeffbeta = 0 for all\nbetaindex.\n\n\\textbf{Remarks:}\nThe first two solutions apply directly over any field of characteristic zero.\n(The result fails in characteristic $p>0$ because we may take\nnonpolyn = (fixedone^2 + \\cdots + fixedlast^2)^p = fixedone^{2p} + \\cdots + fixedlast^{2p}.)\nThe third solution can be extended to complex coefficients\nby replacing $nonpolyn(\\nabla)$ by its complex conjugate, and again the result\nmay be deduced for any field of characteristic zero.\nStanley also suggests\nSection 5 of the arXiv e-print \\texttt{math.CO/0502363} for\nsome  algebraic background for this problem."
    },
    "garbled_string": {
      "map": {
        "x": "zqyplmnaf",
        "x_1": "vhotjkswie",
        "x_n": "piygksewcx",
        "x_i": "ralxgovmte",
        "P": "utdmweklsi",
        "Q": "zbiodnwteh",
        "R": "gxyrpvclai",
        "m": "qlhjsevbru",
        "n": "ocwdqtemyl",
        "d": "wzyhrulkqa",
        "i": "sfeyovumkl",
        "j": "hifdmrqspt",
        "P_d": "uvourlejks",
        "H_d": "bgikrclyze",
        "c_\\alpha": "mqrktsewdu",
        "\\alpha": "\\nhwzopcvta"
      },
      "question": "Let $utdmweklsi(vhotjkswie,\\dots,piygksewcx)$ denote a polynomial with real coefficients in the\nvariables $vhotjkswie, \\dots, piygksewcx$, and suppose that\n\\[\n\\left( \\frac{\\partial^2}{\\partial vhotjkswie^2} + \\cdots + \\frac{\\partial^2}{\\partial\npiygksewcx^2}\\right) utdmweklsi(vhotjkswie, \\dots,piygksewcx) = 0 \\quad \\mbox{(identically)}\n\\]\nand that\n\\[\nvhotjkswie^2 + \\cdots + piygksewcx^2 \\mbox{ divides } utdmweklsi(vhotjkswie, \\dots, piygksewcx).\n\\]\nShow that $utdmweklsi=0$ identically.",
      "solution": "\\textbf{First solution:}\nPut $zbiodnwteh = vhotjkswie^2 + \\cdots + piygksewcx^2$. Since $zbiodnwteh$ is homogeneous, $utdmweklsi$ is divisible\nby $zbiodnwteh$ if and only if each of the homogeneous components of $utdmweklsi$ is divisible\nby $zbiodnwteh$. It is thus sufficient to solve the problem in case $utdmweklsi$ itself is\nhomogeneous, say of degree $wzyhrulkqa$.\n\nSuppose that we have a factorization $utdmweklsi = zbiodnwteh^{qlhjsevbru} gxyrpvclai$ for some $qlhjsevbru>0$, where\n$gxyrpvclai$ is homogeneous\nof degree $wzyhrulkqa$ and not divisible by $zbiodnwteh$;\nnote that the homogeneity implies that\n\\[\n\\sum_{sfeyovumkl=1}^{ocwdqtemyl} ralxgovmte \\frac{\\partial gxyrpvclai}{\\partial ralxgovmte} = wzyhrulkqa gxyrpvclai.\n\\]\nWrite $\\nabla^2$ as shorthand for $\\frac{\\partial^2}{\\partial\nvhotjkswie^2} + \\cdots + \\frac{\\partial^2}{\\partial piygksewcx^2}$; then\n\\begin{align*}\n0 &= \\nabla^2 utdmweklsi \\\\\n&= 2 qlhjsevbru ocwdqtemyl zbiodnwteh^{qlhjsevbru-1}gxyrpvclai + zbiodnwteh^{qlhjsevbru} \\nabla^2 gxyrpvclai + 2 \\sum_{sfeyovumkl=1}^{ocwdqtemyl} 2 qlhjsevbru ralxgovmte zbiodnwteh^{qlhjsevbru-1}\n\\frac{\\partial gxyrpvclai}{\\partial\nralxgovmte} \\\\\n&= zbiodnwteh^{qlhjsevbru} \\nabla^2 gxyrpvclai + (2 qlhjsevbru ocwdqtemyl + 4 qlhjsevbru wzyhrulkqa) zbiodnwteh^{qlhjsevbru-1} gxyrpvclai.\n\\end{align*}\nSince $qlhjsevbru>0$, this forces $gxyrpvclai$ to be divisible by $zbiodnwteh$, contradiction.\n\n\\textbf{Second solution:}\n(by Noam Elkies)\nRetain notation as in the first solution.\nLet $uvourlejks$ be the set of homogeneous\npolynomials of degree $wzyhrulkqa$, and let $bgikrclyze$ be the subset of $uvourlejks$\nof polynomials killed by $\\nabla^2$, which has dimension\n$\\geq \\dim(uvourlejks) - \\dim(uvourlejks_{wzyhrulkqa-2})$; the given problem amounts to showing\nthat this inequality is actually an equality.\n\nConsider the operator $zbiodnwteh \\nabla^2$ (i.e., apply $\\nabla^2$ then multiply\nby $zbiodnwteh$) on $uvourlejks$; its zero eigenspace is precisely $bgikrclyze$.\nBy the calculation from the first solution, if $gxyrpvclai \\in uvourlejks$, then\n\\[\n\\nabla^2 (zbiodnwteh gxyrpvclai) - zbiodnwteh \\nabla^2 gxyrpvclai = (2 ocwdqtemyl + 4 wzyhrulkqa) gxyrpvclai.\n\\]\nConsequently, $zbiodnwteh^{hifdmrqspt} bgikrclyze_{wzyhrulkqa-2 hifdmrqspt}$ is contained in the eigenspace of $zbiodnwteh \\nabla^2$\non $uvourlejks$ of eigenvalue\n\\[\n(2 ocwdqtemyl + 4(wzyhrulkqa-2 hifdmrqspt)) + \\cdots + (2 ocwdqtemyl + 4(wzyhrulkqa-2)).\n\\]\nIn particular, the $zbiodnwteh^{hifdmrqspt} bgikrclyze_{wzyhrulkqa-2 hifdmrqspt}$ lie in distinct eigenspaces, so are\nlinearly independent within $uvourlejks$. But by dimension counting,\ntheir total dimension is at least that of $uvourlejks$.\nHence they exhaust $uvourlejks$, and the zero eigenspace cannot have dimension\ngreater than $\\dim(uvourlejks) - \\dim(uvourlejks_{wzyhrulkqa-2})$, as desired.\n\n\\textbf{Third solution:}\n(by Richard Stanley)\nWrite $zqyplmnaf = (vhotjkswie, \\dots, piygksewcx)$ and $\\nabla = (\\frac{\\partial}{\\partial vhotjkswie},\n\\dots, \\frac{\\partial}{\\partial piygksewcx})$.\nSuppose that $utdmweklsi(zqyplmnaf) = zbiodnwteh(zqyplmnaf)(vhotjkswie^2 + \\cdots + piygksewcx^2)$. Then\n\\[\nutdmweklsi(\\nabla)utdmweklsi(zqyplmnaf) = zbiodnwteh(\\nabla)(\\nabla^2)utdmweklsi(zqyplmnaf) =0.\n\\]\nOn the other hand,\nif $utdmweklsi(zqyplmnaf) = \\sum_{\\nhwzopcvta} mqrktsewdu zqyplmnaf^{\\nhwzopcvta}$ (where $\\nhwzopcvta = (\\nhwzopcvta_1,\n\\dots, \\nhwzopcvta_{ocwdqtemyl})$ and $zqyplmnaf^{\\nhwzopcvta} = vhotjkswie^{\\nhwzopcvta_1} \\cdots\npiygksewcx^{\\nhwzopcvta_{ocwdqtemyl}}$), then the constant term of $utdmweklsi(\\nabla)utdmweklsi(zqyplmnaf)$\nis seen to be $\\sum_{\\nhwzopcvta} mqrktsewdu^2$. Hence $mqrktsewdu = 0$ for all\n$\\nhwzopcvta$.\n\n\\textbf{Remarks:}\nThe first two solutions apply directly over any field of characteristic zero.\n(The result fails in characteristic $p>0$ because we may take\nutdmweklsi = (vhotjkswie^2 + \\cdots + piygksewcx^2)^p = vhotjkswie^{2p} + \\cdots + piygksewcx^{2p}.)\nThe third solution can be extended to complex coefficients\nby replacing utdmweklsi(\\nabla) by its complex conjugate, and again the result\nmay be deduced for any field of characteristic zero.\nStanley also suggests\nSection 5 of the arXiv e-print \\texttt{math.CO/0502363} for\nsome  algebraic background for this problem."
    },
    "kernel_variant": {
      "question": "Let $n\\ge 2$ and $m\\ge 1$ be fixed integers and work in the polynomial ring  \n$\\mathbf C[x_1,\\dots ,x_n]$.  Put  \n\\[\n\\Delta\\;=\\;\\sum_{i=1}^{n}\\frac{\\partial^{2}}{\\partial x_i^{2}},\n\\qquad  \nQ(x_1,\\dots ,x_n)\\;=\\;x_1^{2}+\\cdots +x_n^{2}.\n\\]\n\nFor a polynomial $P_0\\in\\mathbf C[x_1,\\dots ,x_n]$ define the {\\it\nLaplacian chain}\n\\[\nP_k\\;:=\\;\\Delta^{\\,k}P_0\\qquad(k=0,1,\\dots ,m).\n\\]\n\n(a)  (Extinction criterion)  \nAssume that\n\\[\nP_m=0\n\\quad\\text{and}\\quad\nQ^{\\,m}\\;\\hbox{ divides }\\;P_0 .\n\\tag{A}\n\\]\nProve that $P_0$ must be the zero polynomial.\n\n(b)  (Sharpness and structure at the critical exponent)  \nDescribe {\\it completely} the set of non–zero polynomials that satisfy\n\\[\nP_m=0\n\\quad\\text{and}\\quad \nQ^{\\,m-1}\\;\\hbox{ divides }\\;P_0 .\n\\tag{B}\n\\]\nIn particular, show that all such polynomials are of the form\n$P_0 = Q^{\\,m-1}H$ with $H$ harmonic, and compute the dimension of the\nvector space of solutions in terms of $n,\\;m$ and the degree of $H$.\n\n(This shows that the exponent $m$ in part (a) is best possible.)\n\n--------------------------------------------------------------------",
      "solution": "Throughout the proof we let  \n\\[\nE\\;=\\;\\sum_{i=1}^{n}x_i\\frac{\\partial}{\\partial x_i}\n\\qquad\\text{(Euler operator)},\n\\]\nand we frequently use the standard commutator identity  \n\\[\n[\\Delta,Q]\\;=\\;4E+2n.\n\\tag{1}\n\\]\n\nWe also recall the {\\it Fischer decomposition}: for every $D\\ge 0$\n\\[\n\\mathcal P_D\n\\;=\\;\n\\bigoplus_{j=0}^{\\lfloor D/2\\rfloor} Q^{\\,j}\\mathcal H_{D-2j},\n\\tag{2}\n\\]\nwhere $\\mathcal P_D$ is the space of homogeneous polynomials of degree\n$D$ and $\\mathcal H_{D-2j}$ the subspace of homogeneous {\\it harmonic}\npolynomials of degree $D-2j$.\n\n--------------------------------------------------------------------\nPart (a)  (Proof of the extinction criterion)\n\nStep 0.  Reduction to the homogeneous case.  \nWrite $P_0=\\sum_{D\\ge 0}P_0^{(D)}$ with\n$P_0^{(D)}\\in\\mathcal P_D$.\nBecause $\\Delta$ lowers the degree by $2$ and multiplication by\n$Q$ raises it by $2$, the two hypotheses in (A) hold {\\it separately}\nfor every homogeneous degree.\nHence it suffices to treat the case where $P_0$ itself is homogeneous\nof degree $D$.\n\nStep 1.  Locate the smallest power of $Q$ inside $P_0$.  \nInsert $P_0$ into the direct sum (2):\n\\[\nP_0\n\\;=\\;\n\\sum_{j=r}^{\\lfloor D/2\\rfloor} Q^{\\,j}\\,H_{j},\n\\qquad\nH_{j}\\in\\mathcal H_{D-2j},\n\\quad\nH_{r}\\neq 0.\n\\tag{3}\n\\]\nThe smallest $j$ occurring in (3) is denoted $r$.\nThe divisibility $Q^{\\,m}\\mid P_0$ given in (A) implies\n\\[\nr\\;\\ge\\;m.\n\\tag{4}\n\\]\n\nStep 2.  The effect of $\\Delta$ on a summand $Q^{\\,j}H$.  \nLet $H$ be homogeneous {\\it harmonic} of degree $d$.  \nFrom (1) and induction one obtains\n\\[\n\\Delta\\!\\bigl(Q^{\\,j}H\\bigr)\n=\n2j\\bigl(2d+2j+n-2\\bigr)\\;Q^{\\,j-1}H,\n\\qquad\n(j\\ge 1),\n\\tag{5}\n\\]\nwhose coefficient is never $0$ because $j\\ge 1,\\;d\\ge 0,\\;n\\ge 2$.\n\nStep 3.  Propagating the leading $Q$–power through the chain.  \nSuccessive applications of (5) to the decomposition (3) give\n\\[\n\\begin{aligned}\nP_1&=\\Delta P_0\n      =Q^{\\,r-1}(\\hbox{non–zero harmonic})\n      +(\\hbox{higher powers of }Q),\\\\[4pt]\nP_2&=\\Delta P_1\n      =Q^{\\,r-2}(\\hbox{non–zero harmonic})\n      +\\cdots,\\\\\n&\\ \\ \\vdots\\\\\nP_{r}&=\\Delta^{\\,r}P_0\n      =Q^{\\,0}(\\hbox{non–zero harmonic})+\\cdots\n      \\;\\neq\\;0.\n\\end{aligned}\n\\tag{6}\n\\]\nConsequently\n\\[\nP_{r}\\;\\neq\\;0.\n\\tag{7}\n\\]\n\nStep 4.  Contradiction.  \nWrite $r=m+s$ with $s\\ge 0$ by virtue of (4).  \nBecause $P_{k}=\\Delta^{\\,k}P_0$ by definition,  \n\\[\nP_{r}\n=\n\\Delta^{\\,m+s}P_0\n=\n\\Delta^{\\,s}\\bigl(\\Delta^{\\,m}P_0\\bigr)\n=\n\\Delta^{\\,s}P_m.\n\\tag{8}\n\\]\nBut $P_m=0$ by hypothesis (A); therefore the right–hand side of (8) is\n$0$, contradicting (7).  Hence no {\\it non–zero} homogeneous\npolynomial satisfies (A).\n\nStep 5.  Conclusion for the general (inhomogeneous) $P_0$.  \nEvery homogeneous component of $P_0$ must vanish, so $P_0\\equiv 0$.\nThis completes the proof of part (a).  $\\square$\n\n--------------------------------------------------------------------\nPart (b)  (Polynomials satisfying (B))\n\nNecessity.  \nAssume $P_0$ fulfils (B), that is,\n\\[\n\\Delta^{\\,m}P_0=0\n\\quad\\text{and}\\quad\nP_0=Q^{\\,m-1}R\n\\ \\text{ for some }\\\nR\\in\\mathbf C[x_1,\\dots ,x_n].\n\\tag{9}\n\\]\nWrite $R$ as a sum of homogeneous components; as in part (a),\nit suffices to treat the homogeneous case\n$R=R^{(d)}$.  Equation (9) then reads\n\\[\nP_0\\;=\\;Q^{\\,m-1}\\,R^{(d)},\\qquad \\deg R^{(d)}=d.\n\\tag{10}\n\\]\nApply $\\Delta^{\\,m-1}$ to (10) and use (5) repeatedly:\n\\[\n\\Delta^{\\,m-1}P_0\n=\nc\\;R^{(d)},\n\\qquad\nc\\;=\\;\\prod_{j=1}^{m-1}2j\\bigl(2d+2j+n-2\\bigr)\\;\\neq\\;0.\n\\tag{11}\n\\]\nBecause $\\Delta^{\\,m}P_0=0$, the last Laplacian\nsends the left–hand side of (11) to $0$, i.e.\n$\\Delta R^{(d)}=0$.\nThus $R^{(d)}$ is {\\it harmonic},\nand a fortiori every inhomogeneous $R$ occurring in (9)\nis a sum of harmonic components.\n\nSufficiency.  \nConversely, take any harmonic polynomial $H$\n(necessarily $\\Delta H=0$) and put\n\\[\nP_0\\;=\\;Q^{\\,m-1}H.\n\\tag{12}\n\\]\nA single use of (5) shows\n\\[\n\\Delta\\!\\bigl(Q^{\\,m-1}H\\bigr)\n=\nc_1\\,Q^{\\,m-2}H,\\quad\n\\Delta^{\\,2}\\!\\bigl(Q^{\\,m-1}H\\bigr)\n=c_2\\,Q^{\\,m-3}H,\\quad\\dots ,\n\\Delta^{\\,m-1}\\!\\bigl(Q^{\\,m-1}H\\bigr)\n=c_{m-1}\\,H,\n\\tag{13}\n\\]\nwith $c_j\\neq 0$ for all $j$.\nSince $H$ is harmonic, the final Laplacian yields\n$\\Delta^{\\,m}P_0=0$, so (B) is satisfied.\n\nDimension count.  \nLet $\\mathcal H_{d}$ denote the space of homogeneous harmonic\npolynomials of degree $d$; it has well–known dimension\n\\[\n\\dim\\mathcal H_{d}\n=\n\\binom{n+d-1}{d}-\\binom{n+d-3}{d-2}.\n\\tag{14}\n\\]\nFixing $m$ and allowing $d$ to vary, the solution space of (B) is\n\\[\n\\mathcal S_{m,n}\n=\n\\bigoplus_{d\\ge 0} Q^{\\,m-1}\\mathcal H_{d},\n\\]\nhence\n\\[\n\\dim\\mathcal S_{m,n}\n=\n\\sum_{d\\ge 0}\\dim\\mathcal H_{d}\n\\;=\\;\\infty .\n\\]\nIf one restricts the total {\\it degree}\n$\\deg P_0=D$, only the summand with\n$d=D-2(m-1)$ survives; in that graded piece\n\\[\n\\dim\\bigl(\\mathcal S_{m,n}\\cap\\mathcal P_{D}\\bigr)\n=\n\\dim\\mathcal H_{D-2(m-1)}\n=\n\\binom{n+D-2m}{D-2(m-1)}\n-\n\\binom{n+D-2m-2}{D-2m}.\n\\tag{15}\n\\]\n\nWe have thus described every non–zero solution of (B) and\ncomputed its degree–wise dimension, proving the sharpness of\npart (a).  $\\qquad\\square$\n\n--------------------------------------------------------------------",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.795926",
        "was_fixed": false,
        "difficulty_analysis": "1. Multiple iterated constraints:  \n   The original problem involved a \\emph{single} Laplacian constraint\n   ΔP=0 and one divisibility hypothesis.  The enhanced variant imposes an\n   entire \\emph{chain} of differential conditions (Δ^kP=0 up to order m)\n   intertwined with a \\emph{hierarchy} of divisibility requirements,\n   greatly enlarging the logical structure that must be navigated.\n\n2. Necessity of advanced structural tools:  \n   Solving the new task forces the competitor to employ the Fischer\n   decomposition of polynomial representations—an algebraic–analytic\n   result that is absent from elementary treatments.  The commutator\n   identity [Δ,Q]=4E+2n and its iterative consequences are likewise\n   indispensable.\n\n3. Cascading inductive argument:  \n   One has to follow how the \\emph{lowest} Q–power propagates under each\n   application of Δ, then combine this with the highest allowed order of\n   poly-harmonicity.  This cascading reasoning is markedly subtler than\n   the single–step contradiction in the original exercise.\n\n4. Dimensional bookkeeping:  \n   Keeping track of degrees, Q-powers and orders of differentiation\n   simultaneously adds a combinatorial layer that was entirely missing\n   before.\n\n5. No shortcut via pattern recognition:  \n   The solution demands\n   – recognition of the Fischer direct sum,\n   – manipulation of non-trivial commutators,\n   – an inductive descent on the minimal Q–power,  \n   none of which can be guessed by rote or by mimicking the original\n   argument verbatim.\n\nFor these reasons the enhanced variant is substantially more intricate,\nrequiring deeper theoretical insights and several interconnected\ntechniques, thus fully meeting the “significantly harder” criterion."
      }
    },
    "original_kernel_variant": {
      "question": "Let n ≥ 2 and m ≥ 1 be fixed integers and work in the polynomial ring ℂ[x₁,…,xₙ].  \nDenote  \n  Δ  =  ∂²/∂x₁² + ⋯ + ∂²/∂xₙ²  (the n–dimensional Laplacian)  \n  Q(x) = x₁² + ⋯ + xₙ²     (the standard Euclidean quadratic form).\n\nFor a polynomial P₀ ∈ ℂ[x₁,…,xₙ] define the Δ–chain  \n  P_k := Δ^{k}P₀   (k = 0,1,…,m).                                (1)\n\nAssume simultaneously  \n(A)  P_m = 0  (i.e. P₀ is poly-harmonic of order m),  \n(B)  Q^{\\,m−k} divides P_k for every k = 0,1,…,m−1.                (2)\n\nProve that P₀ must be the zero polynomial.\n\n——————————————————————————————————————————————",
      "solution": "We rectify the flaw pointed out in the review while preserving the original line of thought.\n\nStep 0 – Reduction to the homogeneous case.  \nWrite P₀ = Σ_{d≥0} P₀^{(d)} with P₀^{(d)} homogeneous of degree d.  \nBecause Δ lowers the degree by 2 and multiplication by Q raises it by 2, the hypotheses (1)–(2) hold for each homogeneous degree separately.  \nHence it suffices to prove the statement for a non-zero homogeneous P₀ of degree D. From now on P₀ is assumed homogeneous.\n\nStep 1 – Fischer decomposition.  \nLet 𝓗_D be the space of homogeneous harmonic polynomials of degree D.  \nThe classical Fischer decomposition yields  \n  ℙ_D = ⊕_{j=0}^{⌊D/2⌋} Q^{\\,j} 𝓗_{D−2j}.                        (3)\n\nWrite P₀ accordingly as  \n  P₀ = Σ_{j=r}^{⌊D/2⌋} Q^{\\,j}H_{j},  H_{j} ∈ 𝓗_{D−2j},  H_{r} ≠ 0,   (4)  \nwhere r is the smallest exponent of Q occurring in P₀.  \nCondition (2) with k = 0 implies Q^{m} | P₀, hence r ≥ m.            (5)\n\nStep 2 – Action of Δ on Q^j H.  \nFor homogeneous harmonic H of degree d one has the commutator identity  \n  [Δ,Q] = 4E + 2n, E = Σ x_i ∂/∂x_i.                           (6)\n\nUsing (6) or induction one finds, for every j ≥ 1,  \n  Δ(Q^{\\,j}H) = 2j(2d + 2j + n − 2) Q^{\\,j−1} H.                (7)\n\nBecause n ≥ 2, d ≥ 0 and j ≥ 1, the coefficient is non-zero; hence Δ sends Q^{\\,j}H to a non-zero multiple of Q^{\\,j−1}H.\n\nStep 3 – Propagating the leading Q-power through the chain.  \nApplying (7) successively to the expansion (4) gives\n\n  P₁ = ΔP₀ = Q^{\\,r−1}(non-zero harmonic) + (higher Q-powers),  \n  P₂ = ΔP₁ = Q^{\\,r−2}(non-zero harmonic) + ⋯,  \n                     ⋮  \n  P_{r} = Δ^{\\,r}P₀ = Q^{\\,0}(non-zero harmonic) + ⋯ ≠ 0.        (8)\n\nThus  \n  P_r = Δ^{\\,r}P₀ ≠ 0.                                            (9)\n\nStep 4 – Contradiction with the poly-harmonic hypothesis (corrected).  \nBecause of (5) we may write  \n  r = m + s  with  s ≥ 0.                                      (10)\n\nBy definition (1) and (10),\n\n  P_r = Δ^{\\,r}P₀ = Δ^{\\,m+s}P₀ = Δ^{\\,s}(Δ^{\\,m}P₀) = Δ^{\\,s}P_m.   (11)\n\nHypothesis (A) says P_m = 0, hence the right-hand side of (11) is 0, i.e.  \n  P_r = 0.                                                       (12)\n\nBut (9) asserts P_r ≠ 0, a contradiction. Consequently no non-zero homogeneous polynomial can satisfy (A)–(B), and therefore the assumed P₀ is impossible.\n\nStep 5 – Conclusion for the general P₀.  \nEach homogeneous component of P₀ must vanish, so P₀ itself is identically zero. □",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.607559",
        "was_fixed": false,
        "difficulty_analysis": "1. Multiple iterated constraints:  \n   The original problem involved a \\emph{single} Laplacian constraint\n   ΔP=0 and one divisibility hypothesis.  The enhanced variant imposes an\n   entire \\emph{chain} of differential conditions (Δ^kP=0 up to order m)\n   intertwined with a \\emph{hierarchy} of divisibility requirements,\n   greatly enlarging the logical structure that must be navigated.\n\n2. Necessity of advanced structural tools:  \n   Solving the new task forces the competitor to employ the Fischer\n   decomposition of polynomial representations—an algebraic–analytic\n   result that is absent from elementary treatments.  The commutator\n   identity [Δ,Q]=4E+2n and its iterative consequences are likewise\n   indispensable.\n\n3. Cascading inductive argument:  \n   One has to follow how the \\emph{lowest} Q–power propagates under each\n   application of Δ, then combine this with the highest allowed order of\n   poly-harmonicity.  This cascading reasoning is markedly subtler than\n   the single–step contradiction in the original exercise.\n\n4. Dimensional bookkeeping:  \n   Keeping track of degrees, Q-powers and orders of differentiation\n   simultaneously adds a combinatorial layer that was entirely missing\n   before.\n\n5. No shortcut via pattern recognition:  \n   The solution demands\n   – recognition of the Fischer direct sum,\n   – manipulation of non-trivial commutators,\n   – an inductive descent on the minimal Q–power,  \n   none of which can be guessed by rote or by mimicking the original\n   argument verbatim.\n\nFor these reasons the enhanced variant is substantially more intricate,\nrequiring deeper theoretical insights and several interconnected\ntechniques, thus fully meeting the “significantly harder” criterion."
      }
    }
  },
  "checked": true,
  "problem_type": "proof",
  "iteratively_fixed": true
}