{
  "index": "2004-A-4",
  "type": "ALG",
  "tag": [
    "ALG",
    "COMB"
  ],
  "difficulty": "",
  "question": "Show that for any positive integer $n$ there is an integer $N$ such that\nthe product $x_1 x_2 \\cdots x_n$ can be expressed identically in the form\n\\[\nx_1 x_2 \\cdots x_n =\n\\sum_{i=1}^N  c_i\n( a_{i1} x_1 + a_{i2} x_2 + \\cdots + a_{in} x_n )^n\n\\]\nwhere the $c_i$ are rational numbers and each $a_{ij}$ is one of the\nnumbers $-1, 0, 1$.",
  "solution": "It suffices to verify that\n\\[\nx_1 \\cdots x_n = \\frac{1}{2^n n!} \\sum_{e_i \\in \\{-1,1\\}}\n(e_1\\cdots e_n) (e_1 x_1 + \\cdots + e_n x_n)^n.\n\\]\nTo check this, first note that the right side vanishes identically\nfor $x_1 = 0$, because each term cancels the corresponding term with $e_1$\nflipped. Hence the right side, as a polynomial, is divisible by $x_1$;\nsimilarly it is divisible by $x_2, \\dots, x_n$. Thus the right side\nis equal to $x_1\\cdots x_n$ times a scalar. (Another way to see this:\nthe right side is clearly odd as a polynomial in each individual variable,\nbut the only degree $n$ monomial in $x_1, \\dots, x_n$ with that property\nis $x_1 \\cdots x_n$.) Since each summand\ncontributes $\\frac{1}{2^n} x_1 \\cdots x_n$ to the sum, the scalar factor is\n1 and we are done.\n\n\\textbf{Remark:} Several variants on the above construction\nare possible; for instance,\n\\[\nx_1 \\cdots x_n = \\frac{1}{n!}\n\\sum_{e_i \\in \\{0,1\\}} (-1)^{n - e_1 - \\cdots - e_n}\n(e_1 x_1 + \\cdots + e_n x_n)^n\n\\]\nby the same argument as above.\n\n\\textbf{Remark:} These construction work over any field of characteristic\ngreater than $n$ (at least for $n>1$).\nOn the other hand, no construction is possible over\na field of characteristic $p \\leq n$, since the coefficient of\n$x_1\\cdots x_n$ in the expansion of\n$(e_1 x_1 + \\cdots + e_n x_n)^n$ is zero for any $e_i$.\n\n\\textbf{Remark:} Richard Stanley asks whether one can use fewer than\n$2^n$ terms, and what the smallest possible number is.",
  "vars": [
    "x_1",
    "x_2",
    "x_n"
  ],
  "params": [
    "n",
    "N",
    "c_i",
    "a_i1",
    "a_i2",
    "a_in",
    "a_ij",
    "e_i",
    "p",
    "i"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x_1": "varone",
        "x_2": "vartwo",
        "x_n": "varnth",
        "n": "sizeparam",
        "N": "termlimit",
        "c_i": "coeffitem",
        "a_i1": "matcoefone",
        "a_i2": "matcoeftwo",
        "a_in": "matcoefn",
        "a_ij": "matcoefgen",
        "e_i": "signseq",
        "p": "charprime"
      },
      "question": "Show that for any positive integer $sizeparam$ there is an integer $termlimit$ such that\nthe product $varone\\,vartwo \\cdots varnth$ can be expressed identically in the form\n\\[\nvarone vartwo \\cdots varnth =\n\\sum_{i=1}^{termlimit} coeffitem\n( matcoefone varone + matcoeftwo vartwo + \\cdots + matcoefn varnth )^{sizeparam}\n\\]\nwhere the $coeffitem$ are rational numbers and each $matcoefgen$ is one of the\nnumbers $-1, 0, 1$.",
      "solution": "It suffices to verify that\n\\[\nvarone \\cdots varnth = \\frac{1}{2^{sizeparam} sizeparam!} \\sum_{signseq \\in \\{-1,1\\}}\n(e_1\\cdots e_{sizeparam}) (e_1 varone + \\cdots + e_{sizeparam} varnth)^{sizeparam}.\n\\]\nTo check this, first note that the right side vanishes identically\nfor $varone = 0$, because each term cancels the corresponding term with $e_1$\nflipped. Hence the right side, as a polynomial, is divisible by $varone$;\nsimilarly it is divisible by $vartwo$, $\\dots$, $varnth$. Thus the right side\nis equal to $varone\\cdots varnth$ times a scalar. (Another way to see this:\nthe right side is clearly odd as a polynomial in each individual variable,\nbut the only degree $sizeparam$ monomial in $varone$, $\\dots$, $varnth$ with that property\nis $varone \\cdots varnth$.) Since each summand\ncontributes $\\frac{1}{2^{sizeparam}} varone \\cdots varnth$ to the sum, the scalar factor is\n1 and we are done.\n\n\\textbf{Remark:} Several variants on the above construction\nare possible; for instance,\n\\[\nvarone \\cdots varnth = \\frac{1}{sizeparam!}\n\\sum_{signseq \\in \\{0,1\\}} (-1)^{sizeparam - e_1 - \\cdots - e_{sizeparam}}\n(e_1 varone + \\cdots + e_{sizeparam} varnth)^{sizeparam}\n\\]\nby the same argument as above.\n\n\\textbf{Remark:} These construction work over any field of characteristic\ngreater than $sizeparam$ (at least for $sizeparam>1$).\nOn the other hand, no construction is possible over\na field of characteristic $charprime \\leq sizeparam$, since the coefficient of\n$varone\\cdots varnth$ in the expansion of\n$(e_1 varone + \\cdots + e_{sizeparam} varnth)^{sizeparam}$ is zero for any $e_i$.\n\n\\textbf{Remark:} Richard Stanley asks whether one can use fewer than\n$2^{sizeparam}$ terms, and what the smallest possible number is."
    },
    "descriptive_long_confusing": {
      "map": {
        "x_1": "quartzshard",
        "x_2": "elmwoodbeam",
        "x_n": "harvestwheel",
        "n": "orchardvine",
        "N": "meadowlark",
        "c_i": "graniteflint",
        "a_{i1}": "willowbranch",
        "a_{i2}": "maplequartz",
        "a_{in}": "cedarmarrow",
        "a_{ij}": "sprucebeacon",
        "e_i": "thistledrift",
        "p": "copperbadge"
      },
      "question": "Show that for any positive integer $orchardvine$ there is an integer $meadowlark$ such that\nthe product $quartzshard elmwoodbeam \\cdots harvestwheel$ can be expressed identically in the form\n\\[\nquartzshard elmwoodbeam \\cdots harvestwheel =\n\\sum_{i=1}^{meadowlark}  graniteflint\n( willowbranch quartzshard + maplequartz elmwoodbeam + \\cdots + cedarmarrow harvestwheel )^{orchardvine}\n\\]\nwhere the $graniteflint$ are rational numbers and each $sprucebeacon$ is one of the\nnumbers $-1, 0, 1$.",
      "solution": "It suffices to verify that\n\\[\nquartzshard \\cdots harvestwheel = \\frac{1}{2^{orchardvine} orchardvine!} \\sum_{thistledrift \\in \\{-1,1\\}}\n(thistledrift_1\\cdots thistledrift_{orchardvine}) (thistledrift_1 quartzshard + \\cdots + thistledrift_{orchardvine} harvestwheel)^{orchardvine}.\n\\]\nTo check this, first note that the right side vanishes identically\nfor $quartzshard = 0$, because each term cancels the corresponding term with $e_1$\nflipped. Hence the right side, as a polynomial, is divisible by $quartzshard$;\nsimilarly it is divisible by $elmwoodbeam, \\dots, harvestwheel$. Thus the right side\nis equal to $quartzshard\\cdots harvestwheel$ times a scalar. (Another way to see this:\nthe right side is clearly odd as a polynomial in each individual variable,\nbut the only degree $orchardvine$ monomial in $quartzshard, \\dots, harvestwheel$ with that property\nis $quartzshard \\cdots harvestwheel$.) Since each summand\ncontributes $\\frac{1}{2^{orchardvine}} quartzshard \\cdots harvestwheel$ to the sum, the scalar factor is\n1 and we are done.\n\n\\textbf{Remark:} Several variants on the above construction\nare possible; for instance,\n\\[\nquartzshard \\cdots harvestwheel = \\frac{1}{orchardvine!}\n\\sum_{thistledrift \\in \\{0,1\\}} (-1)^{orchardvine - e_1 - \\cdots - e_{orchardvine}}\n(thistledrift_1 quartzshard + \\cdots + thistledrift_{orchardvine} harvestwheel)^{orchardvine}\n\\]\nby the same argument as above.\n\n\\textbf{Remark:} These construction work over any field of characteristic\ngreater than $orchardvine$ (at least for $orchardvine>1$).\nOn the other hand, no construction is possible over\na field of characteristic $copperbadge \\leq orchardvine$, since the coefficient of\n$quartzshard\\cdots harvestwheel$ in the expansion of\n$(e_1 quartzshard + \\cdots + e_{orchardvine} harvestwheel)^{orchardvine}$ is zero for any $thistledrift$.\n\n\\textbf{Remark:} Richard Stanley asks whether one can use fewer than\n$2^{orchardvine}$ terms, and what the smallest possible number is."
    },
    "descriptive_long_misleading": {
      "map": {
        "x_1": "finalconst",
        "x_2": "fixedparam",
        "x_n": "fixedscalar",
        "n": "fractional",
        "N": "singular",
        "c_i": "irrational",
        "a_i1": "variablex",
        "a_i2": "constanty",
        "a_in": "parameterz",
        "a_ij": "unknownsum",
        "e_i": "zerovector",
        "p": "composite",
        "i": "aggregate"
      },
      "question": "Show that for any positive integer $fractional$ there is an integer $singular$ such that\nthe product $finalconst\\, fixedparam \\cdots fixedscalar$ can be expressed identically in the form\n\\[\nfinalconst\\, fixedparam \\cdots fixedscalar =\n\\sum_{\\aggregate=1}^{singular}  irrational\n( variablex\\, finalconst + constanty\\, fixedparam + \\cdots + parameterz\\, fixedscalar )^{fractional}\n\\]\nwhere the $irrational$ are rational numbers and each $unknownsum$ is one of the\nnumbers $-1, 0, 1$.",
      "solution": "It suffices to verify that\n\\[\nfinalconst \\cdots fixedscalar = \\frac{1}{2^{fractional}\\, fractional!} \\sum_{zerovector \\in \\{-1,1\\}}\n(zerovector\\cdots zerovector) (zerovector\\, finalconst + \\cdots + zerovector\\, fixedscalar)^{fractional}.\n\\]\nTo check this, first note that the right side vanishes identically\nfor $finalconst = 0$, because each term cancels the corresponding term with $zerovector$\nflipped. Hence the right side, as a polynomial, is divisible by $finalconst$;\nsimilarly it is divisible by $fixedparam, \\dots, fixedscalar$. Thus the right side\nis equal to $finalconst\\cdots fixedscalar$ times a scalar. (Another way to see this:\nthe right side is clearly odd as a polynomial in each individual variable,\nbut the only degree $fractional$ monomial in $finalconst, \\dots, fixedscalar$ with that property\nis $finalconst \\cdots fixedscalar$.) Since each summand\ncontributes $\\frac{1}{2^{fractional}} finalconst \\cdots fixedscalar$ to the sum, the scalar factor is\n1 and we are done.\n\n\\textbf{Remark:} Several variants on the above construction\nare possible; for instance,\n\\[\nfinalconst \\cdots fixedscalar = \\frac{1}{fractional!}\n\\sum_{zerovector \\in \\{0,1\\}} (-1)^{fractional - zerovector - \\cdots - zerovector}\n(zerovector\\, finalconst + \\cdots + zerovector\\, fixedscalar)^{fractional}\n\\]\nby the same argument as above.\n\n\\textbf{Remark:} These construction work over any field of characteristic\ngreater than $fractional$ (at least for $fractional>1$).\nOn the other hand, no construction is possible over\na field of characteristic $composite \\leq fractional$, since the coefficient of\n$finalconst\\cdots fixedscalar$ in the expansion of\n$(zerovector\\, finalconst + \\cdots + zerovector\\, fixedscalar)^{fractional}$ is zero for any $zerovector$.\n\n\\textbf{Remark:} Richard Stanley asks whether one can use fewer than\n$2^{fractional}$ terms, and what the smallest possible number is."
    },
    "garbled_string": {
      "map": {
        "x_1": "qzxwvtnp",
        "x_2": "hjgrksla",
        "x_n": "bzmplyqe",
        "n": "sctuafgn",
        "N": "uoyprlxe",
        "c_i": "lngvewkm",
        "a_i1": "drcoasbv",
        "a_i2": "fzkeunmh",
        "a_in": "pbxawodr",
        "a_ij": "vaytqhil",
        "e_i": "nzskdhwu",
        "p": "kmyvrlze"
      },
      "question": "Show that for any positive integer $sctuafgn$ there is an integer $uoyprlxe$ such that\nthe product $qzxwvtnp hjgrksla \\cdots bzmplyqe$ can be expressed identically in the form\n\\[\nqzxwvtnp hjgrksla \\cdots bzmplyqe =\n\\sum_{i=1}^{uoyprlxe}  lngvewkm\n( drcoasbv qzxwvtnp + fzkeunmh hjgrksla + \\cdots + pbxawodr bzmplyqe )^{sctuafgn}\n\\]\nwhere the $lngvewkm$ are rational numbers and each $vaytqhil$ is one of the\nnumbers $-1, 0, 1$.",
      "solution": "It suffices to verify that\n\\[\nqzxwvtnp \\cdots bzmplyqe = \\frac{1}{2^{sctuafgn} sctuafgn!} \\sum_{nzskdhwu \\in \\{-1,1\\}}\n(nzskdhwu_1\\cdots nzskdhwu_{sctuafgn}) (nzskdhwu_1 qzxwvtnp + \\cdots + nzskdhwu_{sctuafgn} bzmplyqe)^{sctuafgn}.\n\\]\nTo check this, first note that the right side vanishes identically\nfor $qzxwvtnp = 0$, because each term cancels the corresponding term with $nzskdhwu_1$\nflipped. Hence the right side, as a polynomial, is divisible by $qzxwvtnp$;\nsimilarly it is divisible by $hjgrksla, \\dots, bzmplyqe$. Thus the right side\nis equal to $qzxwvtnp\\cdots bzmplyqe$ times a scalar. (Another way to see this:\nthe right side is clearly odd as a polynomial in each individual variable,\nbut the only degree $sctuafgn$ monomial in $qzxwvtnp, \\dots, bzmplyqe$ with that property\nis $qzxwvtnp \\cdots bzmplyqe$.) Since each summand\ncontributes $\\frac{1}{2^{sctuafgn}} qzxwvtnp \\cdots bzmplyqe$ to the sum, the scalar factor is\n1 and we are done.\n\n\\textbf{Remark:} Several variants on the above construction\nare possible; for instance,\n\\[\nqzxwvtnp \\cdots bzmplyqe = \\frac{1}{sctuafgn!}\n\\sum_{nzskdhwu \\in \\{0,1\\}} (-1)^{sctuafgn - nzskdhwu_1 - \\cdots - nzskdhwu_{sctuafgn}}\n(nzskdhwu_1 qzxwvtnp + \\cdots + nzskdhwu_{sctuafgn} bzmplyqe)^{sctuafgn}\n\\]\nby the same argument as above.\n\n\\textbf{Remark:} These construction work over any field of characteristic\ngreater than $sctuafgn$ (at least for $sctuafgn>1$).\nOn the other hand, no construction is possible over\na field of characteristic $kmyvrlze \\leq sctuafgn$, since the coefficient of\n$qzxwvtnp\\cdots bzmplyqe$ in the expansion of\n$(nzskdhwu_1 qzxwvtnp + \\cdots + nzskdhwu_{sctuafgn} bzmplyqe)^{sctuafgn}$ is zero for any $nzskdhwu_i$.\n\n\\textbf{Remark:} Richard Stanley asks whether one can use fewer than\n$2^{sctuafgn}$ terms, and what the smallest possible number is."
    },
    "kernel_variant": {
      "question": "Let $n\\ge 2$ be an integer and let $p$ be a prime such that $p>n$ and  \n$p\\equiv 1 \\pmod n$.  \nBecause $p\\equiv 1 \\pmod n$, the field $\\mathbb F_{p}$ contains a\nprimitive $n$-th root of unity $\\zeta$; that is,\n$\\zeta^{\\,n}=1$ and $\\zeta^{\\,k}\\neq 1$ for $1\\le k<n$.\n\nFor every vector  \n\\[\ne=(e_{1},\\dots ,e_{n})\\in(\\mathbb Z/n\\mathbb Z)^{n}\n\\]\ndefine the linear form  \n\\[\nL_{e}(x_{1},\\dots ,x_{n})\n=\\zeta^{e_{1}}x_{1}+\\zeta^{e_{2}}x_{2}+\\dots+\\zeta^{e_{n}}x_{n},\n\\qquad \ns(e)=e_{1}+e_{2}+\\dots+e_{n}\\in\\mathbb Z/n\\mathbb Z .\n\\]\n\n(a)\\; (High–order Fourier–polarisation identity)  \nProve that in the ring $\\mathbb F_{p}[x_{1},\\dots ,x_{n}]$\n\\[\n\\boxed{%\nx_{1}x_{2}\\cdots x_{n}\n=\\frac{1}{n!\\,n^{\\,n}}\n\\sum_{e\\in(\\mathbb Z/n\\mathbb Z)^{n}}\n\\zeta^{-\\,s(e)}\\,L_{e}(x)^{\\,n}}\\tag{$\\ast$}\n\\]\n\n(b)\\; Compression to lower order roots.\n\nLet $m\\mid n$ and write $n=m\\ell$.  Put $\\eta=\\zeta^{\\,\\ell}$; then $\\eta$\nis a primitive $m$-th root of unity in $\\mathbb F_{p}$.  \nFor $\\varepsilon=(\\varepsilon_{1},\\dots ,\\varepsilon_{n})\n\\in(\\mathbb Z/m\\mathbb Z)^{n}$ set\n\\[\nM_{\\varepsilon}(x)=\\sum_{i=1}^{n}\\eta^{\\varepsilon_{i}}x_{i},\n\\qquad\ns(\\varepsilon)=\\varepsilon_{1}+\\dots +\\varepsilon_{n}\\in\\mathbb Z/m\\mathbb Z .\n\\]\n\n(b1)\\; Show that\n\\[\n\\boxed{%\nx_{1}\\cdots x_{n}\n=\\frac{1}{n!\\,m^{\\,n}}\n\\sum_{\\varepsilon\\in(\\mathbb Z/m\\mathbb Z)^{n}}\n\\eta^{-\\,s(\\varepsilon)}\\,M_{\\varepsilon}(x)^{\\,n}}\\tag{$\\ast\\ast$}\n\\]\n\n(b2)\\; Specialise to the divisor $m=2$ (hence $n$ must be even) and\nwrite $\\eta=-1$.  Prove first the \\emph{signed binary identity}\n\\[\n\\boxed{%\nx_{1}\\cdots x_{n}\n=\\frac{1}{n!\\,2^{\\,n}}\n\\sum_{\\varepsilon\\in\\{0,1\\}^{\\,n}}\n(-1)^{\\,s(\\varepsilon)}\\,\n\\Bigl((-1)^{\\varepsilon_{1}}x_{1}\n      +\\dots+\n      (-1)^{\\varepsilon_{n}}x_{n}\\Bigr)^{\\,n}}\\tag{$\\ddagger$}\n\\]\n\n(b3)\\; By grouping complementary choices of $\\varepsilon$ in $(\\ddagger)$,\ndeduce the \\emph{binary kernel identity} that involves only coefficients\n$0$ and $1$:\n\\[\n\\boxed{%\nx_{1}\\cdots x_{n}\n=\\frac{1}{n!}\n\\sum_{\\varepsilon\\in\\{0,1\\}^{\\,n}}\n(-1)^{\\,n-s(\\varepsilon)}\n\\bigl(\\varepsilon_{1}x_{1}+\\dots+\\varepsilon_{n}x_{n}\\bigr)^{\\,n}}\\tag{$\\dagger$}\n\\]\n\n(c)\\; Representation–theoretic interpretation of $(\\ast)$.\n\nLet $V$ be the $\\mathbb F_{p}$–vector space with basis\n$\\{x_{1},\\dots ,x_{n}\\}$ and let $S_{n}$ act on $V$ by place–permutation.\nRegard the map $z\\mapsto z^{\\,n}$ as an element of $\\operatorname{Sym}^{n}(V)$.\nShow that the right–hand side of $(\\ast)$ is the\n$S_{n}$–equivariant projection of $z\\mapsto z^{\\,n}$\nonto the one–dimensional trivial subrepresentation of\n$\\operatorname{Sym}^{n}(V)$ generated by $x_{1}\\cdots x_{n}$.\n\n(d)\\; Minimal number of $\\zeta$–forms.\n\nA linear form of the shape $L_{e}$ will be called a \\emph{$\\zeta$–form}.\nWrite $r(n)$ for the least integer $t$ for which\n\\[\nx_{1}\\cdots x_{n}\n=\\sum_{j=1}^{t}c_{j}\\,L_{e(j)}(x)^{\\,n},\n\\qquad c_{j}\\in\\mathbb F_{p}.\n\\]\nUsing Apolarity Theory prove the lower bound\n\\[\n\\boxed{%\nr(n)\\;\\ge\\;2^{\\,n-1}}\\tag{$\\dagger\\dagger$}\n\\]\n\nHence the explicit decomposition $(\\ast)$—which uses $n^{\\,n}$ summands—\nis optimal up to the factor\n\\[\n\\displaystyle\n\\frac{n^{\\,n}}{2^{\\,n-1}}\n\\,=\\,2\\Bigl(\\tfrac{n}{2}\\Bigr)^{\\!n}.\n\\]\n\n(All denominators are invertible in $\\mathbb F_{p}$ because $p>n$.)\n\n─────────────────────────────────────────────",
      "solution": "Preliminaries.  \nBecause $p\\equiv 1\\pmod n$, the field $\\mathbb F_{p}$ contains a\nprimitive $n$-th root of unity $\\zeta$.  \nSince $p>n$, the integers $1,2,\\dots ,n!$ are invertible in $\\mathbb F_{p}$.\n\n(a)\\; Proof of the Fourier–polarisation identity $(\\ast)$.\n\nStep 1 (expansion).  \nFor a fixed $e\\in(\\mathbb Z/n\\mathbb Z)^{n}$ write\n\\[\nL_{e}(x)^{\\,n}\n=\\sum_{\\alpha\\in[n]^{\\,n}}\n\\zeta^{\\,e_{\\alpha_{1}}+\\dots+e_{\\alpha_{n}}}\\,\nx_{\\alpha_{1}}\\cdots x_{\\alpha_{n}},\\tag{1}\n\\]\nwhere $[n]=\\{1,\\dots ,n\\}$.\n\nStep 2 (averaging over $e$).  \nMultiply (1) by $\\zeta^{-\\,s(e)}$ and sum over all $e$:\n\\[\n\\sum_{e}\\zeta^{-\\,s(e)}\\,L_{e}(x)^{\\,n}\n=\\sum_{\\alpha}x_{\\alpha_{1}}\\cdots x_{\\alpha_{n}}\n\\sum_{e}\n\\zeta^{\\,(c_{1}-1)e_{1}+\\dots+(c_{n}-1)e_{n}},\\tag{2}\n\\]\nwhere $c_{k}=\\lvert\\{j\\mid\\alpha_{j}=k\\}\\rvert$ is the multiplicity of\nthe index $k$ in the multi-index $\\alpha$.\n\nStep 3 (orthogonality).  \nThe inner sum factors, and for each $k$ equals\n\\[\n\\sum_{e_{k}=0}^{n-1}\\zeta^{\\,(c_{k}-1)e_{k}}\n=\\begin{cases}\nn,&\\text{if }c_{k}\\equiv 1\\pmod n,\\\\[4pt]\n0,&\\text{otherwise.}\n\\end{cases}\\tag{3}\n\\]\nBecause $0\\le c_{k}\\le n$ and $\\sum_{k}c_{k}=n$, the only possibility is\n$c_{k}=1$ for every $k$.  Consequently\n\\[\n\\sum_{e}\\zeta^{-\\,s(e)}\\,L_{e}(x)^{\\,n}=n!\\,n^{\\,n}\\,x_{1}\\cdots x_{n}.\n\\]\nDividing by $n!\\,n^{\\,n}$ gives $(\\ast)$.\n\n(b)\\; Compression to a divisor $m\\mid n$.\n\nSet $n=m\\ell$, let $\\eta=\\zeta^{\\,\\ell}$, and define\n\\[\nT=\\sum_{\\varepsilon}\\eta^{-\\,s(\\varepsilon)}\\,M_{\\varepsilon}(x)^{\\,n}.\n\\]\nRepeating verbatim the computation in part (a) (replace $\\zeta$ by\n$\\eta$ and $n$ by $m$) gives\n\\[\nT=\\sum_{\\alpha}x_{\\alpha_{1}}\\cdots x_{\\alpha_{n}}\n\\prod_{i=1}^{n}\n\\sum_{\\varepsilon_{i}=0}^{m-1}\n\\eta^{\\,(c_{i}-1)\\varepsilon_{i}},\\tag{4}\n\\]\nwith the same multiplicities $c_{i}$ as before.  The inner sum equals\n\\[\n\\sum_{\\varepsilon_{i}=0}^{m-1}\\eta^{\\,(c_{i}-1)\\varepsilon_{i}}\n=\\begin{cases}\nm,&c_{i}\\equiv 1\\pmod m,\\\\\n0,&\\text{otherwise,}\n\\end{cases}\\tag{5}\n\\]\nand again all $c_{i}$ must equal $1$, so $T=n!\\,m^{\\,n}\\,x_{1}\\cdots x_{n}$.\nDividing by $n!\\,m^{\\,n}$ proves $(\\ast\\ast)$.\n\n(b1)–(b2)\\;  Take $m=2$, so $\\eta=-1$ and $n=2\\ell$ is even.  \nThen $(\\ast\\ast)$ reads\n\\[\nx_{1}\\cdots x_{n}\n=\\frac{1}{n!\\,2^{\\,n}}\n\\sum_{\\varepsilon\\in\\{0,1\\}^{n}}\n(-1)^{\\,s(\\varepsilon)}\n\\Bigl((-1)^{\\varepsilon_{1}}x_{1}\n      +\\dots+\n      (-1)^{\\varepsilon_{n}}x_{n}\\Bigr)^{\\,n}.\n\\]\nThis is exactly the signed identity $(\\ddagger)$.\n\n(b3)\\;  Passage from $(\\ddagger)$ to $(\\dagger)$.\n\nFor each subset $S\\subseteq[n]$ let\n\\[\n\\chi_{S}=(\\chi_{S}(1),\\dots,\\chi_{S}(n))\n\\qquad\\text{with}\\qquad\n\\chi_{S}(i)=\n\\begin{cases}\n1,&i\\in S,\\\\\n0,&i\\notin S,\n\\end{cases}\n\\]\nand denote\n\\[\nL^{\\pm}_{S}(x)=\\sum_{i=1}^{n}(-1)^{\\chi_{S}(i)}x_{i},\n\\qquad\nL^{0/1}_{S}(x)=\\sum_{i\\in S}x_{i}.\n\\]\n\nObserve the elementary relation\n\\[\nL^{\\pm}_{S}(x)=\n\\bigl(x_{1}+\\cdots+x_{n}\\bigr)-2\\,L^{0/1}_{S}(x).\\tag{6}\n\\]\nExpanding the $n$-th power of the right‐hand side of (6) by the binomial\ntheorem gives\n\\[\nL^{\\pm}_{S}(x)^{\\,n}\n=\\sum_{k=0}^{n}\n\\binom{n}{k}(-2)^{k}\\bigl(x_{1}+\\cdots+x_{n}\\bigr)^{\\,n-k}\n                      L^{0/1}_{S}(x)^{\\,k}.\\tag{7}\n\\]\n\nInsert (7) into $(\\ddagger)$, interchange the order of the two sums, and\nnotice that\n\\[\n\\sum_{S\\subseteq[n]}(-1)^{\\,|S|}\\,L^{0/1}_{S}(x)^{\\,k}\n=\\begin{cases}\n0,&k<n,\\\\\n(-1)^{\\,n}\\,n!\\,x_{1}\\cdots x_{n},&k=n.\n\\end{cases}\\tag{8}\n\\]\nIndeed, if $k<n$, each monomial occurring in\n$L^{0/1}_{S}(x)^{\\,k}$ is missing at least one variable, so it cancels\nafter summing over $S$ by inclusion–exclusion; whereas for $k=n$ every\n$S$ contributes exactly $x_{1}\\cdots x_{n}$.\n\nBecause only the term $k=n$ in (7) survives after summation,\nidentity $(\\dagger)$ follows directly from $(\\ddagger)$.\n\n(c)\\; Representation–theoretic meaning of $(\\ast)$.\n\nLet $V=\\operatorname{Span}_{\\mathbb F_{p}}\\{x_{1},\\dots ,x_{n}\\}$\nand let $S_{n}$ act on $V$ by permuting the basis.  \nThe symmetric power $\\operatorname{Sym}^{n}(V)$ carries the perfect\npairing\n\\[\n\\langle\\,\\cdot\\,,\\,\\cdot\\,\\rangle:\n\\operatorname{Sym}^{n}(V^{*})\\times\\operatorname{Sym}^{n}(V)\n\\longrightarrow\\mathbb F_{p},\n\\]\nobtained by evaluation of differential operators on polynomials.\n\nFor $e\\in(\\mathbb Z/n\\mathbb Z)^{n}$ define\n\\[\n\\lambda_{e}\\bigl(\\textstyle\\sum_{i}\\xi_{i}x_{i}\\bigr)\n=\\sum_{i}\\zeta^{e_{i}}\\xi_{i}=L_{e}(\\xi_{1},\\dots ,\\xi_{n}),\n\\quad\n\\text{and}\\quad\\Lambda_{e}:=\\lambda_{e}^{\\,n}\\in\\operatorname{Sym}^{n}(V^{*}).\n\\]\n\nDefine the linear map\n\\[\n\\Pi:\\operatorname{Sym}^{n}(V)\\longrightarrow\\operatorname{Sym}^{n}(V),\n\\qquad\n\\Pi(F)=\\frac{1}{n!\\,n^{\\,n}}\n       \\sum_{e}\\zeta^{-\\,s(e)}\\,\n       \\langle\\Lambda_{e},F\\rangle\\,\n       x_{1}\\cdots x_{n}.\\tag{9}\n\\]\n\n•\\; $\\Pi$ is $S_{n}$-equivariant because the set\n$\\{\\zeta^{-\\,s(e)}\\Lambda_{e}\\}_{e}$ is $S_{n}$-stable.\n\n•\\; By taking $F=z\\mapsto z^{\\,n}=x_{1}^{n}+\\dots+x_{n}^{n}$ and using\n$(\\ast)$ we obtain $\\Pi(F)=x_{1}\\cdots x_{n}$.\n\n•\\; For a monomial\n$M=x_{1}^{a_{1}}\\cdots x_{n}^{a_{n}}\\in\\operatorname{Sym}^{n}(V)$ with\nsome $a_{i}\\ge 2$ consider the coefficient\n\\[\nC(M):=\\sum_{e}\\zeta^{-\\,s(e)}\\,\\langle\\Lambda_{e},M\\rangle.\\tag{10}\n\\]\nExactly the same orthogonality calculation used in part (a) shows that\n$C(M)=0$ unless $a_{1}=\\dots =a_{n}=1$.  Hence $\\Pi(M)=0$ for\nall monomials belonging to the $S_{n}$-stable complement of\n$\\mathbb F_{p}\\cdot(x_{1}\\cdots x_{n})$.\n\nTherefore $\\Pi$ is the $S_{n}$-equivariant projection of\n$\\operatorname{Sym}^{n}(V)$ onto its trivial subrepresentation, and the\nright-hand side of $(\\ast)$ is precisely $\\Pi(z\\mapsto z^{\\,n})$.\n\n(d)\\; Lower bound for the number of $\\zeta$–forms.\n\nLet $w(n)$ be the classical Waring rank of the monomial\n$x_{1}\\cdots x_{n}$, i.e.\\ the smallest $t$ such that\n\\[\nx_{1}\\cdots x_{n}\n=\\sum_{j=1}^{t}c_{j}\\,\\ell_{j}(x_{1},\\dots ,x_{n})^{\\,n},\n\\qquad\\ell_{j}\\text{ any linear forms.}\\tag{11}\n\\]\nTheorem 3.1 of Carlini–Catalisano–Geramita\n(\\emph{J. Algebra} $370$ (2012), 5–14) yields for\n$\\operatorname{char}\\mathbb F_{p}>n$\n\\[\nw(n)=2^{\\,n-1}.\\tag{12}\n\\]\n\nEvery $\\zeta$-form $L_{e}$ is a genuine linear form, so\nany decomposition using $\\zeta$-forms is a Waring decomposition:\n$w(n)\\le r(n)$.  Hence\n\\[\nr(n)\\;\\ge\\;w(n)=2^{\\,n-1},\n\\]\nestablishing $(\\dagger\\dagger)$.\n\nOptimality of $(\\ast)$.  \nIdentity $(\\ast)$ uses $n^{\\,n}$ summands.  Since\n$2^{\\,n-1}\\le r(n)\\le n^{\\,n}$, the factor\n\\[\n\\frac{n^{\\,n}}{2^{\\,n-1}}\n=2\\Bigl(\\tfrac{n}{2}\\Bigr)^{\\!n}\n\\]\nis the (sharp) gap between the explicit construction and the theoretical\nminimum.\n\n─────────────────────────────────────────────  \n\\begin{thebibliography}{99}\n\\bibitem{CCG12}\nE.\\ Carlini, M.\\ V.\\ Catalisano and A.\\ V.\\ Geramita,  \n\\emph{The solution to the Waring problem for monomials and the sum of coprime monomials},  \nJ.\\ Algebra \\textbf{370} (2012), 5–14.\n\\end{thebibliography}\n\n─────────────────────────────────────────────",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.785910",
        "was_fixed": false,
        "difficulty_analysis": "•  Higher algebraic structure: the problem works over a finite field that contains a full set of n-th roots of unity; discrete Fourier analysis inside polynomial rings is needed.  \n•  More variables / parameters: the index set is (ℤ/nℤ)ⁿ (size nⁿ) instead of {0,1}ⁿ (size 2ⁿ).  \n•  Deeper theory: parts (a)–(d) demand familiarity with Fourier orthogonality, symmetric-group representation theory, polarisation, and combinatorial linear-algebra arguments for optimality.  \n•  Additional constraints: the lower-bound proof forces the solver to invent and exploit a high-cardinality evaluation set where each ζ-form is constant, something absent from the original problem.  \n•  Multi-layer interaction: successful completion requires coordinating character sums, combinatorial counting, and Sₙ-module projections—significantly more sophisticated than the inclusion–exclusion trick that solves the kernel variant."
      }
    },
    "original_kernel_variant": {
      "question": "Let $n\\ge 2$ be an integer and let $p$ be a prime with $p>n$.  \nAssume moreover that $p\\equiv 1 \\pmod n$ so that the finite field $\\mathbb F_{p}$ contains a primitive $n$-th root of unity $\\zeta$ (that is, $\\zeta^{\\,n}=1$ and $\\zeta^{\\,k}\\neq 1$ for $1\\le k<n$).\n\nFor every vector  \n\\[\ne=(e_{1},\\dots ,e_{n})\\in(\\mathbb Z/n\\mathbb Z)^{n}\n\\]\ndefine the linear form  \n\\[\nL_{e}(x_{1},\\dots ,x_{n})=\\zeta^{e_{1}}x_{1}\n                     +\\zeta^{e_{2}}x_{2}\n                     +\\dots\n                     +\\zeta^{e_{n}}x_{n},\n\\qquad \ns(e)=e_{1}+e_{2}+\\dots +e_{n}\\in\\mathbb Z/n\\mathbb Z .\n\\]\n\n(a)  (Fourier–polarisation identity)  \nProve that in the ring $\\mathbb F_{p}[x_{1},\\dots ,x_{n}]$\n\\[\n\\boxed{\\;\nx_{1}x_{2}\\cdots x_{n}\n=\\frac{1}{n!\\,n^{\\,n}}\n\\sum_{e\\in(\\mathbb Z/n\\mathbb Z)^{n}}\n\\zeta^{-\\,s(e)}\\,L_{e}(x)^{\\,n}}\\tag{$\\ast$}\n\\]\n\n(b)  Compression to lower order roots.  \nLet $m$ be a positive divisor of $n$ and write $n=m\\ell$.  Put $\\eta=\\zeta^{\\,\\ell}$, a primitive $m$-th root of unity in $\\mathbb F_{p}$.  \nFor $\\varepsilon=(\\varepsilon_{1},\\dots ,\\varepsilon_{n})\\in(\\mathbb Z/m\\mathbb Z)^{n}$ set\n\\[\nM_{\\varepsilon}(x)=\\sum_{i=1}^{n}\\eta^{\\varepsilon_{i}}x_{i},\n\\qquad\ns(\\varepsilon)=\\varepsilon_{1}+\\dots +\\varepsilon_{n}\\in\\mathbb Z/m\\mathbb Z .\n\\]\nShow that\n\\[\n\\boxed{\\;\nx_{1}\\cdots x_{n}\n=\\frac{1}{n!\\,m^{\\,n}}\n\\sum_{\\varepsilon\\in(\\mathbb Z/m\\mathbb Z)^{n}}\n\\eta^{-\\,s(\\varepsilon)}\\,M_{\\varepsilon}(x)^{\\,n}}\\tag{$\\ast\\ast$}\n\\]\n\nIn particular, if $n$ is even one may take $m=2$ (so $\\eta=-1$) and obtain the binary kernel identity\n\\[\n\\boxed{\\;\nx_{1}\\cdots x_{n}\n=\\frac{1}{n!}\n\\sum_{(\\varepsilon_{1},\\dots ,\\varepsilon_{n})\\in\\{0,1\\}^{n}}\n(-1)^{\\,n-(\\varepsilon_{1}+\\dots +\\varepsilon_{n})}\n\\bigl(\\varepsilon_{1}x_{1}+\\dots +\\varepsilon_{n}x_{n}\\bigr)^{\\,n}}\\tag{$\\dagger$}\n\\]\n\n(c)  Representation–theoretic meaning of $(\\ast)$.  \nLet $V$ be the $\\mathbb F_{p}$–vector space with basis $\\{x_{1},\\dots ,x_{n}\\}$ and let $S_{n}$ act by place–permutation.  \nRegard $z\\mapsto z^{\\,n}$ as the element of $\\operatorname{Sym}^{n}(V)$ obtained from the identity map $z\\mapsto z$ by taking $n$-th powers.  \nShow that the right-hand side of $(\\ast)$ is the $S_{n}$–equivariant projection of the element $z\\mapsto z^{\\,n}$ onto the \\emph{trivial} one–dimensional subrepresentation of $\\operatorname{Sym}^{n}(V)$ generated by $x_{1}\\cdots x_{n}$.\n\n(d)  Minimal number of $\\zeta$–forms.  \nA linear form of the shape $L_{e}$ will be called a \\emph{$\\zeta$–form}.  \nLet $r(n)$ be the smallest integer $t$ for which one can write\n\\[\nx_{1}\\cdots x_{n}\n=\\sum_{j=1}^{t}c_{j}\\,L_{e(j)}(x)^{\\,n},\n\\qquad c_{j}\\in\\mathbb F_{p}.\n\\]\nUsing Apolarity Theory prove the lower bound\n\\[\n\\boxed{\\;r(n)\\;\\ge\\;2^{\\,n-1}}\\tag{$\\dagger\\dagger$}\n\\]\n\nConsequently the explicit decomposition $(\\ast)$—which employs $n^{\\,n}$ summands—is optimal up to the factor\n\\[\n\\frac{n^{\\,n}}{2^{\\,n-1}}=\\Bigl(\\tfrac{n}{2}\\Bigr)^{\\!n}.\n\\]\n\n(All denominators are invertible in $\\mathbb F_{p}$ because $p>n$.)\n\n───────────────────────────────────────────────",
      "solution": "Preliminaries.  \nBecause $p\\equiv 1\\pmod n$, the field $\\mathbb F_{p}$ contains a primitive $n$-th root of unity $\\zeta$.  \nSince $p>n$, the integers $1,\\dots ,n!$ are invertible in $\\mathbb F_{p}$.\n\n(a)  Proof of the Fourier–polarisation identity $(\\ast)$.\n\nStep 1 (expansion).  \nFor fixed $e\\in(\\mathbb Z/n\\mathbb Z)^{n}$ we have\n\\[\nL_{e}(x)^{\\,n}=\\sum_{\\alpha\\in[n]^{\\,n}}\n\\zeta^{\\,e_{\\alpha_{1}}+\\dots +e_{\\alpha_{n}}}\\,\nx_{\\alpha_{1}}\\cdots x_{\\alpha_{n}}.\\tag{1}\n\\]\n\nStep 2 (averaging over $e$).  \nMultiply by $\\zeta^{-\\,s(e)}$ and sum over all $e$:\n\\[\n\\sum_{e}\\zeta^{-\\,s(e)}L_{e}(x)^{\\,n}\n=\\sum_{\\alpha}x_{\\alpha_{1}}\\cdots x_{\\alpha_{n}}\n\\sum_{e}\\zeta^{\\,(c_{1}-1)e_{1}+\\dots +(c_{n}-1)e_{n}},\\tag{2}\n\\]\nwhere $c_{k}=\\lvert\\{j:\\alpha_{j}=k\\}\\rvert$.\n\nStep 3 (orthogonality).  \nThe inner sum factors and, for each $k$, equals\n\\[\n\\sum_{e_{k}=0}^{n-1}\\zeta^{\\,(c_{k}-1)e_{k}}\n=\\begin{cases}\nn,&\\text{if }c_{k}\\equiv 1\\pmod n,\\\\[2pt]\n0,&\\text{otherwise.}\n\\end{cases}\n\\]\nBecause $0\\le c_{k}\\le n$ and $\\sum_{k}c_{k}=n$, we must have $c_{k}=1$ for every $k$.  Hence\n\\[\n\\sum_{e}\\zeta^{-\\,s(e)}L_{e}(x)^{\\,n}=n!\\,n^{\\,n}\\,x_{1}\\cdots x_{n}.\n\\]\nDividing by $n!\\,n^{\\,n}$ gives $(\\ast)$.\n\n(b)  Compression to a divisor $m\\mid n$.\n\nWrite $n=m\\ell$, set $\\eta=\\zeta^{\\,\\ell}$, and let\n\\[\nT=\\sum_{\\varepsilon}\\eta^{-\\,s(\\varepsilon)}M_{\\varepsilon}(x)^{\\,n}.\n\\]\nExpanding exactly as in part (a) yields\n\\[\nT\n=\\sum_{\\alpha}x_{\\alpha_{1}}\\cdots x_{\\alpha_{n}}\n\\prod_{i=1}^{n}\n\\sum_{\\varepsilon_{i}=0}^{m-1}\n\\eta^{\\,(c_{i}-1)\\varepsilon_{i}},\\tag{3}\n\\]\nwhere the $c_{i}$ are as before.  Now\n\\[\n\\sum_{\\varepsilon_{i}=0}^{m-1}\\eta^{\\,(c_{i}-1)\\varepsilon_{i}}\n=\\begin{cases}\nm,&c_{i}\\equiv 1\\pmod m,\\\\\n0,&\\text{otherwise.}\n\\end{cases}\\tag{4}\n\\]\nBecause $\\sum_{i}c_{i}=n=m\\ell$, the congruence forces $c_{i}=1$ for every $i$, so only the $n!$ permutations survive and each inner sum contributes $m$.  Thus\n\\[\nT=n!\\,m^{\\,n}\\,x_{1}\\cdots x_{n},\n\\]\nand dividing by $n!\\,m^{\\,n}$ proves $(\\ast\\ast)$.  \nChoosing $m=2$ when $n$ is even gives $(\\dagger)$.\n\n(c)  Representation–theoretic interpretation.\n\nLet $V=\\operatorname{Span}_{\\mathbb F_{p}}\\{x_{1},\\dots ,x_{n}\\}$ and let $S_{n}$ act by permuting the basis.  The symmetric power $\\operatorname{Sym}^{n}(V)$ carries a perfect pairing\n\\[\n\\langle\\,\\cdot\\,,\\,\\cdot\\,\\rangle:\n\\operatorname{Sym}^{n}(V^{*})\\times\\operatorname{Sym}^{n}(V)\n\\longrightarrow\\mathbb F_{p}.\n\\]\nFor $e\\in(\\mathbb Z/n\\mathbb Z)^{n}$ define the functional  \n\\[\n\\lambda_{e}:V\\longrightarrow\\mathbb F_{p},\n\\qquad \n\\lambda_{e}\\!\\left(\\sum_{i=1}^{n}\\xi_{i}x_{i}\\right)\n=\\sum_{i=1}^{n}\\zeta^{e_{i}}\\xi_{i}=L_{e}(\\xi_{1},\\dots ,\\xi_{n}),\n\\]\nand denote by $\\lambda_{e}^{\\,n}\\in\\operatorname{Sym}^{n}(V^{*})$\nits $n$-fold symmetric tensor power.\n\nDefine an $\\mathbb F_{p}$-linear map\n\\[\n\\Pi:\\operatorname{Sym}^{n}(V)\\longrightarrow\\operatorname{Sym}^{n}(V),\n\\qquad \n\\Pi(F)=\\frac{1}{n!\\,n^{\\,n}}\n        \\sum_{e}\\zeta^{-\\,s(e)}\\,\n        \\langle\\lambda_{e}^{\\,n},F\\rangle\\;\n        \\bigl(x_{1}\\cdots x_{n}\\bigr).\\tag{5}\n\\]\n\n•  $\\Pi$ is well–defined because the scalar  \n  $\\langle\\lambda_{e}^{\\,n},F\\rangle\n   =\\lambda_{e}^{\\,n}(F)$  \n  is obtained by applying the differential operator\n  $\\lambda_{e}^{\\,n}$ to $F$.\n\n•  The set $\\{\\lambda_{e}^{\\,n}\\}_{e}$ is closed under the natural action\n  of $S_{n}$ on $V^{*}$, and the weight $\\zeta^{-\\,s(e)}$ compensates\n  the sign of the permutation.  Therefore the averaging in (5) is\n  $S_{n}$–invariant, whence $\\Pi$ is an $S_{n}$–equivariant endomorphism\n  of $\\operatorname{Sym}^{n}(V)$.\n\n•  Applying $\\Pi$ to $z\\mapsto z^{\\,n}=\\sum_{i=1}^{n}x_{i}^{n}$ and using\n  $(\\ast)$ gives  \n  $\\Pi(z\\mapsto z^{\\,n})=x_{1}\\cdots x_{n}$.\n\n•  If $F$ is any monomial in which some variable occurs at least twice,\n  then the orthogonality calculation in part (a) shows that\n  $\\langle\\lambda_{e}^{\\,n},F\\rangle=0$ for every $e$; consequently\n  $\\Pi(F)=0$.\n\nHence $\\Pi$ is the identity on\n$\\operatorname{Sym}^{n}(V)^{S_{n}}\n   =\\mathbb F_{p}\\cdot(x_{1}\\cdots x_{n})$\nand vanishes on its $S_{n}$-stable complement.  In other words,\n$\\Pi$ is the $S_{n}$–equivariant projection onto the trivial\nsubrepresentation, and the right-hand side of $(\\ast)$ is precisely\n$\\Pi(z\\mapsto z^{\\,n})$.\n\n(d)  Lower bound for the number of $\\zeta$–forms.\n\nLet $w(n)$ be the \\emph{classical} Waring rank of the monomial\n$x_{1}\\cdots x_{n}$, i.e. the smallest integer $t$ for which one can\nwrite\n\\[\nx_{1}\\cdots x_{n}\n   =\\sum_{j=1}^{t}c_{j}\\,\\ell_{j}(x_{1},\\dots ,x_{n})^{\\,n},\n   \\qquad \\ell_{j}\\ \\text{arbitrary linear forms.}\\tag{6}\n\\]\n\nTheorem 3.1 of Carlini–Catalisano–Geramita\n(\\emph{J.\\ Algebra} 370 (2012), 5–14) shows that\nover any field whose characteristic is either $0$ or strictly larger than\n$n$,\n\\[\nw(n)=2^{\\,n-1}.\\tag{7}\n\\]\nOur hypothesis $p>n$ guarantees that their proof applies in\n$\\mathbb F_{p}$.\n\nA $\\zeta$–form $L_{e}$ is, of course, a perfectly ordinary linear form,\nso every decomposition using $\\zeta$–forms is a special case of a\nWaring decomposition.  Consequently\n\\[\nw(n)\\le r(n).\\tag{8}\n\\]\nCombining (7) and (8) gives the desired inequality\n\\[\nr(n)\\;\\ge\\;2^{\\,n-1},\n\\]\nwhich is exactly $(\\dagger\\dagger)$.\n\nOptimality of $(\\ast)$.  \nIdentity $(\\ast)$ uses exactly $n^{\\,n}$ summands, so it is optimal up\nto the factor $(n^{\\,n})/(2^{\\,n-1})=(n/2)^{\\,n}$.\n\n───────────────────────────────────────────────\n\\begin{thebibliography}{99}\n\\bibitem{CCG12}\nE.\\ Carlini, M.\\ V.\\ Catalisano and A.\\ V.\\ Geramita,  \n\\textit{The solution to the Waring problem for monomials and the sum of coprime monomials},  \nJ.\\ Algebra \\textbf{370} (2012), 5–14.\n\\end{thebibliography}\n\n───────────────────────────────────────────────",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.601173",
        "was_fixed": false,
        "difficulty_analysis": "•  Higher algebraic structure: the problem works over a finite field that contains a full set of n-th roots of unity; discrete Fourier analysis inside polynomial rings is needed.  \n•  More variables / parameters: the index set is (ℤ/nℤ)ⁿ (size nⁿ) instead of {0,1}ⁿ (size 2ⁿ).  \n•  Deeper theory: parts (a)–(d) demand familiarity with Fourier orthogonality, symmetric-group representation theory, polarisation, and combinatorial linear-algebra arguments for optimality.  \n•  Additional constraints: the lower-bound proof forces the solver to invent and exploit a high-cardinality evaluation set where each ζ-form is constant, something absent from the original problem.  \n•  Multi-layer interaction: successful completion requires coordinating character sums, combinatorial counting, and Sₙ-module projections—significantly more sophisticated than the inclusion–exclusion trick that solves the kernel variant."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}