{
  "index": "1940-A-6",
  "type": "ALG",
  "tag": [
    "ALG"
  ],
  "difficulty": "",
  "question": "6. \\( f(x) \\) is a polynomial of degree \\( n \\), such that a power of \\( f(x) \\) is divisible by a power of its derivative \\( f^{\\prime}(x) \\); i.e., \\( [f(x)]^{p} \\) is divisible by \\( \\left[f^{\\prime}(x)\\right]^{q} ; p, q \\), positive integers. Prove that \\( f(x) \\) is divisible by \\( f^{\\prime}(x) \\) and that \\( f(x) \\) has a single root of multiplicity \\( n \\).",
  "solution": "Solution. Let the factorization of \\( f \\) be\n\\[\nf=\\alpha p_{1}{ }_{1}^{e_{1}} p_{2}{ }^{e_{2}} \\cdots p_{k}{ }^{e k}\n\\]\nwhere \\( \\alpha \\) is a scalar and \\( p_{1}, p_{2}, \\ldots, p_{k} \\) are distinct monic irreducible polynomials, and \\( e_{1}, e_{2}, \\ldots, e_{k} \\) are positive integers. Using the product rule for differentiation,\n\\[\nf^{\\prime}=\\alpha \\sum_{i=1}^{k} e_{i} p_{1}^{e_{1}} \\cdots p_{i-1}^{e_{i-1}} p_{i}^{e_{i}-1} p_{i+1}^{e_{i+1}} \\cdots p_{k^{e k}} \\cdot p_{i}{ }^{\\prime} .\n\\]\n\nSince \\( \\boldsymbol{p}_{j^{e j-1}} \\) divides each term of this sum and \\( \\boldsymbol{p}_{j}{ }^{e_{j}} \\) divides all terms but one which it definitely does not divide, one sees that\n\\[\nf^{\\prime}=p_{1}^{e_{1}-1} p_{2}^{e_{2}-1} \\cdots p_{k}^{e k-1} \\cdot g\n\\]\nwhere \\( g \\) is a polynomial not divisible by any of the \\( p_{i} \\) 's. Since some power of \\( f \\) is divisible by a power of \\( f^{\\prime} \\), any irreducible factor of \\( g \\) divides a power of \\( f \\). By the unique factorization theorem for polynomials this is impossible, so one concludes that \\( g \\) has no irreducible factors; hence\n\\[\ng \\text { has degree } 0 .\n\\]\n\nNow the degree of \\( f \\) is \\( n=\\sum_{i=1}^{k} e_{i} d_{i} \\) where \\( d_{i} \\) is the degree of \\( p_{i} \\). By (1) and (2) the degree of \\( f^{\\prime} \\) is \\( n-1=\\sum_{i=1}^{k}\\left(e_{i}-1\\right) d_{i} \\), and subtracting we obtain \\( 1=\\sum_{i=1}^{k} d_{i} \\). Since the \\( d \\) 's are positive integers, we conclude that \\( k=1, d_{1}=1 \\). Hence\n\\[\nf=\\alpha p_{1}{ }^{e_{1}}=\\alpha p_{1}{ }^{n}\n\\]\nwhere \\( p_{t} \\) is a linear polynomial. Therefore \\( f \\) has a simple root of multiplicity \\( n \\), and \\( f \\) is indeed divisible by \\( f^{\\prime}=n \\alpha p_{1}{ }^{n-1} \\).\n\nThe proof could be made less formalistic if we assume that the polynomials are defined over the complex field. We could then start off with\n\\[\nf(x)=\\alpha\\left(x-a_{1}\\right)^{e_{1}}\\left(x-a_{2}\\right)^{e_{2}} \\cdots\\left(x-a_{k}\\right)^{e_{k}} .\n\\]\n\nRemark. The proof just given uses the ring of formal polynomials over an arbitrary field, with differentiation being formally defined. The proof remains valid for arbitrary fields of characteristic zero (this hypothesis is used at the point where it is asserted that there is a term in the sum not divisible by \\( p_{i}{ }^{e}{ }^{e} \\). The result is false for fields of characteristic \\( q \\neq 0 \\). Consider the polynomial \\( x\\left(x^{q}-1\\right) \\). Its derivative is \\( x^{q}-1 \\) which divides \\( f \\), but \\( f \\) does not have a single root of multiplicity \\( q+1 \\).",
  "vars": [
    "x",
    "f",
    "g",
    "i",
    "j"
  ],
  "params": [
    "n",
    "p",
    "q",
    "k",
    "p_i",
    "e_i",
    "d_i",
    "a_i"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "indepvar",
        "f": "origpoly",
        "g": "helperpoly",
        "i": "indexvar",
        "j": "indexalt",
        "n": "polydgree",
        "p": "powerparam",
        "q": "expoparam",
        "k": "factorcnt",
        "p_i": "factorpoly",
        "e_i": "expoplist",
        "d_i": "degplist",
        "a_i": "rootlist"
      },
      "question": "6. \\( origpoly(indepvar) \\) is a polynomial of degree \\( polydgree \\), such that a power of \\( origpoly(indepvar) \\) is divisible by a power of its derivative \\( origpoly^{\\prime}(indepvar) \\); i.e., \\( [origpoly(indepvar)]^{powerparam} \\) is divisible by \\( [origpoly^{\\prime}(indepvar)]^{expoparam};\\; powerparam, expoparam \\), positive integers. Prove that \\( origpoly(indepvar) \\) is divisible by \\( origpoly^{\\prime}(indepvar) \\) and that \\( origpoly(indepvar) \\) has a single root of multiplicity \\( polydgree \\).",
      "solution": "Solution. Let the factorization of \\( origpoly \\) be\n\\[\norigpoly=\\alpha\\, factorpoly_{1}^{expoplist_{1}}\\, factorpoly_{2}^{expoplist_{2}}\\cdots factorpoly_{factorcnt}^{expoplist_{factorcnt}}\n\\]\nwhere \\( \\alpha \\) is a scalar and \\( factorpoly_{1}, factorpoly_{2}, \\ldots, factorpoly_{factorcnt} \\) are distinct monic irreducible polynomials, and \\( expoplist_{1}, expoplist_{2}, \\ldots, expoplist_{factorcnt} \\) are positive integers. Using the product rule for differentiation,\n\\[\norigpoly^{\\prime}= \\alpha \\sum_{indexvar=1}^{factorcnt} expoplist_{indexvar}\\, factorpoly_{1}^{expoplist_{1}}\\cdots factorpoly_{indexvar-1}^{expoplist_{indexvar-1}} factorpoly_{indexvar}^{expoplist_{indexvar}-1} factorpoly_{indexvar+1}^{expoplist_{indexvar+1}}\\cdots factorpoly_{factorcnt}^{expoplist_{factorcnt}}\\cdot factorpoly_{indexvar}^{\\prime}.\n\\]\n\nSince \\( \\boldsymbol{factorpoly}_{indexalt^{expoplist_{indexalt}-1}} \\) divides each term of this sum and \\( \\boldsymbol{factorpoly}_{indexalt}^{expoplist_{indexalt}} \\) divides all terms but one which it definitely does not divide, one sees that\n\\[\norigpoly^{\\prime}=factorpoly_{1}^{expoplist_{1}-1} factorpoly_{2}^{expoplist_{2}-1}\\cdots factorpoly_{factorcnt}^{expoplist_{factorcnt}-1}\\cdot helperpoly\n\\]\nwhere \\( helperpoly \\) is a polynomial not divisible by any of the \\( factorpoly_{indexvar} \\)'s. Since some power of \\( origpoly \\) is divisible by a power of \\( origpoly^{\\prime} \\), any irreducible factor of \\( helperpoly \\) divides a power of \\( origpoly \\). By the unique factorization theorem for polynomials this is impossible, so one concludes that \\( helperpoly \\) has no irreducible factors; hence\n\\[\nhelperpoly \\text{ has degree } 0 .\n\\]\n\nNow the degree of \\( origpoly \\) is \\( polydgree=\\sum_{indexvar=1}^{factorcnt} expoplist_{indexvar}\\, degplist_{indexvar} \\) where \\( degplist_{indexvar} \\) is the degree of \\( factorpoly_{indexvar} \\). By (1) and (2) the degree of \\( origpoly^{\\prime} \\) is \\( polydgree-1=\\sum_{indexvar=1}^{factorcnt}\\left(expoplist_{indexvar}-1\\right) degplist_{indexvar} \\), and subtracting we obtain \\( 1=\\sum_{indexvar=1}^{factorcnt} degplist_{indexvar} \\). Since the \\( degplist \\)'s are positive integers, we conclude that \\( factorcnt=1,\\; degplist_{1}=1 \\). Hence\n\\[\norigpoly=\\alpha\\, factorpoly_{1}^{expoplist_{1}}=\\alpha\\, factorpoly_{1}^{polydgree}\n\\]\nwhere \\( factorpoly_{t} \\) is a linear polynomial. Therefore \\( origpoly \\) has a simple root of multiplicity \\( polydgree \\), and \\( origpoly \\) is indeed divisible by \\( origpoly^{\\prime}=polydgree \\alpha\\, factorpoly_{1}^{polydgree-1} \\).\n\nThe proof could be made less formalistic if we assume that the polynomials are defined over the complex field. We could then start off with\n\\[\norigpoly(indepvar)=\\alpha\\left(indepvar-rootlist_{1}\\right)^{expoplist_{1}}\\left(indepvar-rootlist_{2}\\right)^{expoplist_{2}}\\cdots\\left(indepvar-rootlist_{factorcnt}\\right)^{expoplist_{factorcnt}} .\n\\]\n\nRemark. The proof just given uses the ring of formal polynomials over an arbitrary field, with differentiation being formally defined. The proof remains valid for arbitrary fields of characteristic zero (this hypothesis is used at the point where it is asserted that there is a term in the sum not divisible by \\( factorpoly_{indexvar}^{expoplist_{indexvar}} \\). The result is false for fields of characteristic \\( expoparam \\neq 0 \\). Consider the polynomial \\( indepvar\\left(indepvar^{expoparam}-1\\right) \\). Its derivative is \\( indepvar^{expoparam}-1 \\) which divides \\( origpoly \\), but \\( origpoly \\) does not have a single root of multiplicity \\( expoparam+1 \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "marblesack",
        "f": "lanterngear",
        "g": "thimblecup",
        "i": "parsleybet",
        "j": "tornadofly",
        "n": "willowstem",
        "p": "riverstone",
        "q": "meadowlark",
        "k": "driftcloud",
        "p_i": "orchardmist",
        "e_i": "crystalvine",
        "d_i": "starlumina",
        "a_i": "cobblestone"
      },
      "question": "6. \\( lanterngear(marblesack) \\) is a polynomial of degree \\( willowstem \\), such that a power of \\( lanterngear(marblesack) \\) is divisible by a power of its derivative \\( lanterngear^{\\prime}(marblesack) \\); i.e., \\( [lanterngear(marblesack)]^{riverstone} \\) is divisible by \\( \\left[lanterngear^{\\prime}(marblesack)\\right]^{meadowlark} ; riverstone, meadowlark \\), positive integers. Prove that \\( lanterngear(marblesack) \\) is divisible by \\( lanterngear^{\\prime}(marblesack) \\) and that \\( lanterngear(marblesack) \\) has a single root of multiplicity \\( willowstem \\).",
      "solution": "Solution. Let the factorization of \\( lanterngear \\) be\n\\[\nlanterngear=\\alpha orchardmist_{1}{ }_{1}^{crystalvine_{1}} orchardmist_{2}{ }^{crystalvine_{2}} \\cdots orchardmist_{driftcloud}{ }^{crystalvine driftcloud}\n\\]\nwhere \\( \\alpha \\) is a scalar and \\( orchardmist_{1}, orchardmist_{2}, \\ldots, orchardmist_{driftcloud} \\) are distinct monic irreducible polynomials, and \\( crystalvine_{1}, crystalvine_{2}, \\ldots, crystalvine_{driftcloud} \\) are positive integers. Using the product rule for differentiation,\n\\[\nlanterngear^{\\prime}=\\alpha \\sum_{parsleybet=1}^{driftcloud} crystalvine_{parsleybet} orchardmist_{1}^{crystalvine_{1}} \\cdots orchardmist_{parsleybet-1}^{crystalvine_{parsleybet-1}} orchardmist_{parsleybet}^{crystalvine_{parsleybet}-1} orchardmist_{parsleybet+1}^{crystalvine_{parsleybet+1}} \\cdots orchardmist_{driftcloud^{crystalvine driftcloud}} \\cdot orchardmist_{parsleybet}{ }^{\\prime} .\n\\]\n\nSince \\( \\boldsymbol{orchardmist}_{tornadofly^{crystalvine tornadofly-1}} \\) divides each term of this sum and \\( \\boldsymbol{orchardmist}_{tornadofly}{ }^{crystalvine_{tornadofly}} \\) divides all terms but one which it definitely does not divide, one sees that\n\\[\nlanterngear^{\\prime}=orchardmist_{1}^{crystalvine_{1}-1} orchardmist_{2}^{crystalvine_{2}-1} \\cdots orchardmist_{driftcloud}^{crystalvine driftcloud-1} \\cdot thimblecup\n\\]\nwhere \\( thimblecup \\) is a polynomial not divisible by any of the \\( orchardmist_{parsleybet} \\) 's. Since some power of \\( lanterngear \\) is divisible by a power of \\( lanterngear^{\\prime} \\), any irreducible factor of \\( thimblecup \\) divides a power of \\( lanterngear \\). By the unique factorization theorem for polynomials this is impossible, so one concludes that \\( thimblecup \\) has no irreducible factors; hence\n\\[\nthimblecup \\text { has degree } 0 .\n\\]\n\nNow the degree of \\( lanterngear \\) is \\( willowstem=\\sum_{parsleybet=1}^{driftcloud} crystalvine_{parsleybet} starlumina_{parsleybet} \\) where \\( starlumina_{parsleybet} \\) is the degree of \\( orchardmist_{parsleybet} \\). By (1) and (2) the degree of \\( lanterngear^{\\prime} \\) is \\( willowstem-1=\\sum_{parsleybet=1}^{driftcloud}\\left(crystalvine_{parsleybet}-1\\right) starlumina_{parsleybet} \\), and subtracting we obtain \\( 1=\\sum_{parsleybet=1}^{driftcloud} starlumina_{parsleybet} \\). Since the \\( starlumina \\) 's are positive integers, we conclude that \\( driftcloud=1, starlumina_{1}=1 \\). Hence\n\\[\nlanterngear=\\alpha orchardmist_{1}{ }^{crystalvine_{1}}=\\alpha orchardmist_{1}{ }^{willowstem}\n\\]\nwhere \\( orchardmist_{t} \\) is a linear polynomial. Therefore \\( lanterngear \\) has a simple root of multiplicity \\( willowstem \\), and \\( lanterngear \\) is indeed divisible by \\( lanterngear^{\\prime}=willowstem \\alpha orchardmist_{1}{ }^{willowstem-1} \\).\n\nThe proof could be made less formalistic if we assume that the polynomials are defined over the complex field. We could then start off with\n\\[\nlanterngear(marblesack)=\\alpha\\left(marblesack-cobblestone_{1}\\right)^{crystalvine_{1}}\\left(marblesack-cobblestone_{2}\\right)^{crystalvine_{2}} \\cdots\\left(marblesack-cobblestone_{driftcloud}\\right)^{crystalvine_{driftcloud}} .\n\\]\n\nRemark. The proof just given uses the ring of formal polynomials over an arbitrary field, with differentiation being formally defined. The proof remains valid for arbitrary fields of characteristic zero (this hypothesis is used at the point where it is asserted that there is a term in the sum not divisible by \\( orchardmist_{parsleybet}{ }^{crystalvine}{ }^{crystalvine} \\). The result is false for fields of characteristic \\( meadowlark \\neq 0 \\). Consider the polynomial \\( marblesack\\left(marblesack^{meadowlark}-1\\right) \\). Its derivative is \\( marblesack^{meadowlark}-1 \\) which divides \\( lanterngear \\), but \\( lanterngear \\) does not have a single root of multiplicity \\( meadowlark+1 \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "constantval",
        "f": "nonpolynomial",
        "g": "divisiblepoly",
        "i": "finalindex",
        "j": "initialindex",
        "n": "nondegree",
        "p": "rootorder",
        "q": "logorder",
        "k": "singlefactor",
        "p_i": "reducpoly",
        "e_i": "rootvalue",
        "d_i": "orderless",
        "a_i": "coefficient"
      },
      "question": "6. \\( nonpolynomial(constantval) \\) is a polynomial of degree \\( nondegree \\), such that a power of \\( nonpolynomial(constantval) \\) is divisible by a power of its derivative \\( nonpolynomial^{\\prime}(constantval) \\); i.e., \\( [nonpolynomial(constantval)]^{rootorder} \\) is divisible by \\( \\left[nonpolynomial^{\\prime}(constantval)\\right]^{logorder} ; rootorder, logorder \\), positive integers. Prove that \\( nonpolynomial(constantval) \\) is divisible by \\( nonpolynomial^{\\prime}(constantval) \\) and that \\( nonpolynomial(constantval) \\) has a single root of multiplicity \\( nondegree \\).",
      "solution": "Solution. Let the factorization of \\( nonpolynomial \\) be\n\\[\nnonpolynomial=\\alpha reducpoly_{1}^{rootvalue_{1}} reducpoly_{2}^{rootvalue_{2}} \\cdots reducpoly_{singlefactor}^{rootvalue_{singlefactor}}\n\\]\nwhere \\( \\alpha \\) is a scalar and \\( reducpoly_{1}, reducpoly_{2}, \\ldots, reducpoly_{singlefactor} \\) are distinct monic irreducible polynomials, and \\( rootvalue_{1}, rootvalue_{2}, \\ldots, rootvalue_{singlefactor} \\) are positive integers. Using the product rule for differentiation,\n\\[\nnonpolynomial^{\\prime}=\\alpha \\sum_{finalindex=1}^{singlefactor} rootvalue_{finalindex} reducpoly_{1}^{rootvalue_{1}} \\cdots reducpoly_{finalindex-1}^{rootvalue_{finalindex-1}} reducpoly_{finalindex}^{rootvalue_{finalindex}-1} reducpoly_{finalindex+1}^{rootvalue_{finalindex+1}} \\cdots reducpoly_{singlefactor}^{rootvalue_{singlefactor}} \\cdot reducpoly_{finalindex}^{\\prime} .\n\\]\n\nSince \\( \\boldsymbol{reducpoly}_{initialindex^{rootvalue_{initialindex}-1}} \\) divides each term of this sum and \\( \\boldsymbol{reducpoly}_{initialindex}^{rootvalue_{initialindex}} \\) divides all terms but one which it definitely does not divide, one sees that\n\\[\nnonpolynomial^{\\prime}=reducpoly_{1}^{rootvalue_{1}-1} reducpoly_{2}^{rootvalue_{2}-1} \\cdots reducpoly_{singlefactor}^{rootvalue_{singlefactor}-1} \\cdot divisiblepoly\n\\]\nwhere \\( divisiblepoly \\) is a polynomial not divisible by any of the \\( reducpoly_{finalindex} \\)'s. Since some power of \\( nonpolynomial \\) is divisible by a power of \\( nonpolynomial^{\\prime} \\), any irreducible factor of \\( divisiblepoly \\) divides a power of \\( nonpolynomial \\). By the unique factorization theorem for polynomials this is impossible, so one concludes that \\( divisiblepoly \\) has no irreducible factors; hence\n\\[\ndivisiblepoly \\text { has degree } 0 .\n\\]\n\nNow the degree of \\( nonpolynomial \\) is \\( nondegree=\\sum_{finalindex=1}^{singlefactor} rootvalue_{finalindex} \\, orderless_{finalindex} \\) where \\( orderless_{finalindex} \\) is the degree of \\( reducpoly_{finalindex} \\). By (1) and (2) the degree of \\( nonpolynomial^{\\prime} \\) is \\( nondegree-1=\\sum_{finalindex=1}^{singlefactor}\\left(rootvalue_{finalindex}-1\\right) orderless_{finalindex} \\), and subtracting we obtain \\( 1=\\sum_{finalindex=1}^{singlefactor} orderless_{finalindex} \\). Since the \\( orderless \\)'s are positive integers, we conclude that \\( singlefactor=1, orderless_{1}=1 \\). Hence\n\\[\nnonpolynomial=\\alpha reducpoly_{1}^{rootvalue_{1}}=\\alpha reducpoly_{1}^{nondegree}\n\\]\nwhere \\( reducpoly_{1} \\) is a linear polynomial. Therefore \\( nonpolynomial \\) has a simple root of multiplicity \\( nondegree \\), and \\( nonpolynomial \\) is indeed divisible by \\( nonpolynomial^{\\prime}=nondegree \\, \\alpha \\, reducpoly_{1}^{nondegree-1} \\).\n\nThe proof could be made less formalistic if we assume that the polynomials are defined over the complex field. We could then start off with\n\\[\nnonpolynomial(constantval)=\\alpha\\left(constantval-coefficient_{1}\\right)^{rootvalue_{1}}\\left(constantval-coefficient_{2}\\right)^{rootvalue_{2}} \\cdots\\left(constantval-coefficient_{singlefactor}\\right)^{rootvalue_{singlefactor}} .\n\\]\n\nRemark. The proof just given uses the ring of formal polynomials over an arbitrary field, with differentiation being formally defined. The proof remains valid for arbitrary fields of characteristic zero (this hypothesis is used at the point where it is asserted that there is a term in the sum not divisible by \\( reducpoly_{finalindex}^{rootvalue_{finalindex}} \\). The result is false for fields of characteristic \\( logorder \\neq 0 \\). Consider the polynomial \\( constantval\\left(constantval^{logorder}-1\\right) \\). Its derivative is \\( constantval^{logorder}-1 \\) which divides \\( nonpolynomial \\), but \\( nonpolynomial \\) does not have a single root of multiplicity \\( logorder+1 \\)."
    },
    "garbled_string": {
      "map": {
        "x": "zfmqtbsy",
        "f": "hvnogtwa",
        "g": "qplzsmik",
        "i": "oqlxfgmn",
        "j": "vmskqthz",
        "n": "rqyzfjkc",
        "p": "xtwgjald",
        "q": "radjhumi",
        "k": "belqzton",
        "p_i": "qxzlmfob",
        "e_i": "yscrande",
        "d_i": "akvptroe",
        "a_i": "zotswlen"
      },
      "question": "6. \\( hvnogtwa(zfmqtbsy) \\) is a polynomial of degree \\( rqyzfjkc \\), such that a power of \\( hvnogtwa(zfmqtbsy) \\) is divisible by a power of its derivative \\( hvnogtwa^{\\prime}(zfmqtbsy) \\); i.e., \\( [hvnogtwa(zfmqtbsy)]^{xtwgjald} \\) is divisible by \\( \\left[hvnogtwa^{\\prime}(zfmqtbsy)\\right]^{radjhumi} ; xtwgjald, radjhumi \\), positive integers. Prove that \\( hvnogtwa(zfmqtbsy) \\) is divisible by \\( hvnogtwa^{\\prime}(zfmqtbsy) \\) and that \\( hvnogtwa(zfmqtbsy) \\) has a single root of multiplicity \\( rqyzfjkc \\).",
      "solution": "Solution. Let the factorization of \\( hvnogtwa \\) be\n\\[\nhvnogtwa=\\alpha qxzlmfob_{1}^{yscrande_{1}} qxzlmfob_{2}^{yscrande_{2}} \\cdots qxzlmfob_{belqzton}^{yscrande_{belqzton}}\n\\]\nwhere \\( \\alpha \\) is a scalar and \\( qxzlmfob_{1}, qxzlmfob_{2}, \\ldots, qxzlmfob_{belqzton} \\) are distinct monic irreducible polynomials, and \\( yscrande_{1}, yscrande_{2}, \\ldots, yscrande_{belqzton} \\) are positive integers. Using the product rule for differentiation,\n\\[\nhvnogtwa^{\\prime}=\\alpha \\sum_{oqlxfgmn=1}^{belqzton} yscrande_{oqlxfgmn} \nqxzlmfob_{1}^{yscrande_{1}} \\cdots qxzlmfob_{oqlxfgmn-1}^{yscrande_{oqlxfgmn-1}} \nqxzlmfob_{oqlxfgmn}^{yscrande_{oqlxfgmn}-1} \nqxzlmfob_{oqlxfgmn+1}^{yscrande_{oqlxfgmn+1}} \\cdots qxzlmfob_{belqzton}^{yscrande_{belqzton}} \\cdot qxzlmfob_{oqlxfgmn}^{\\prime} .\n\\]\n\nSince \\(\\boldsymbol{qxzlmfob}_{vmskqthz^{yscrande vmskqthz-1}}\\) divides each term of this sum and \\(\\boldsymbol{qxzlmfob}_{vmskqthz}^{yscrande_{vmskqthz}}\\) divides all terms but one which it definitely does not divide, one sees that\n\\[\nhvnogtwa^{\\prime}=qxzlmfob_{1}^{yscrande_{1}-1} qxzlmfob_{2}^{yscrande_{2}-1} \\cdots qxzlmfob_{belqzton}^{yscrande_{belqzton}-1} \\cdot qplzsmik\n\\]\nwhere \\( qplzsmik \\) is a polynomial not divisible by any of the \\( qxzlmfob_{i} \\)'s. Since some power of \\( hvnogtwa \\) is divisible by a power of \\( hvnogtwa^{\\prime} \\), any irreducible factor of \\( qplzsmik \\) divides a power of \\( hvnogtwa \\). By the unique factorization theorem for polynomials this is impossible, so one concludes that \\( qplzsmik \\) has no irreducible factors; hence\n\\[\nqplzsmik \\text { has degree } 0 .\n\\]\n\nNow the degree of \\( hvnogtwa \\) is \\( rqyzfjkc=\\sum_{oqlxfgmn=1}^{belqzton} yscrande_{oqlxfgmn} akvptroe_{oqlxfgmn} \\) where \\( akvptroe_{oqlxfgmn} \\) is the degree of \\( qxzlmfob_{oqlxfgmn} \\). By (1) and (2) the degree of \\( hvnogtwa^{\\prime} \\) is \\( rqyzfjkc-1=\\sum_{oqlxfgmn=1}^{belqzton}\\left(yscrande_{oqlxfgmn}-1\\right) akvptroe_{oqlxfgmn} \\), and subtracting we obtain \\( 1=\\sum_{oqlxfgmn=1}^{belqzton} akvptroe_{oqlxfgmn} \\). Since the \\( akvptroe \\)'s are positive integers, we conclude that \\( belqzton=1, akvptroe_{1}=1 \\). Hence\n\\[\nhvnogtwa=\\alpha qxzlmfob_{1}^{yscrande_{1}}=\\alpha qxzlmfob_{1}^{rqyzfjkc}\n\\]\nwhere \\( qxzlmfob_{1} \\) is a linear polynomial. Therefore \\( hvnogtwa \\) has a simple root of multiplicity \\( rqyzfjkc \\), and \\( hvnogtwa \\) is indeed divisible by \\( hvnogtwa^{\\prime}=rqyzfjkc \\alpha qxzlmfob_{1}^{rqyzfjkc-1} \\).\n\nThe proof could be made less formalistic if we assume that the polynomials are defined over the complex field. We could then start off with\n\\[\nhvnogtwa(zfmqtbsy)=\\alpha\\left(zfmqtbsy-zotswlen_{1}\\right)^{yscrande_{1}}\\left(zfmqtbsy-zotswlen_{2}\\right)^{yscrande_{2}} \\cdots\\left(zfmqtbsy-zotswlen_{belqzton}\\right)^{yscrande_{belqzton}} .\n\\]\n\nRemark. The proof just given uses the ring of formal polynomials over an arbitrary field, with differentiation being formally defined. The proof remains valid for arbitrary fields of characteristic zero (this hypothesis is used at the point where it is asserted that there is a term in the sum not divisible by \\( qxzlmfob_{oqlxfgmn}^{yscrande_{oqlxfgmn}} \\). The result is false for fields of characteristic \\( radjhumi \\neq 0 \\). Consider the polynomial \\( zfmqtbsy\\left(zfmqtbsy^{radjhumi}-1\\right) \\). Its derivative is \\( zfmqtbsy^{radjhumi}-1 \\) which divides \\( hvnogtwa \\), but \\( hvnogtwa \\) does not have a single root of multiplicity \\( radjhumi+1 \\)."
    },
    "kernel_variant": {
      "question": "Let 𝐊 be an algebraically closed field of characteristic 0 and put  \n  R = 𝐊[x₁,…,xₙ] \\;(n ≥ 1).  \n\nFor a non–constant polynomial F∈R of total degree m≥1 set  \n\n  JF := (∂F/∂x₁,…,∂F/∂xₙ)  ⊂ R,  JFˢ := (JF)ˢ\n\n(the ordinary s-th power of the Jacobian ideal).  \nFix positive integers r,s and assume  \n\n  (★)  JFˢ ⊂ (F)ʳ.  \n\nDenote by rad F the radical ∏ gᵢ of the principal ideal (F) (product of the\ndistinct irreducible factors of F) and write  \n\n  F = c·g₁^{e₁}⋯g_k^{e_k} (c∈𝐊*, gᵢ irreducible, eᵢ≥1).  \n\nDefine                                                            \n\n  τ := ⌈ s /(s−r) ⌉ (so τ≥2 and s>r).\n\n1.  Prove that s>r and that every multiplicity eᵢ satisfies the sharp\n  inequality                                                     \n\n    (s−r)·eᵢ  ≥  s      (†)  \n\n  equivalently eᵢ ≥ τ for all i.\n\n2.  Conversely, assume s>r and eᵢ ≥ τ for every irreducible divisor gᵢ of F.\n  Show that (★) holds.\n\n3.  Conclude the following equivalence (classification of (★)):  \n\n  JFˢ ⊂ (F)ʳ ⇔ s>r and F is divisible by (rad F)^{τ}.  \n\n  In words: (★) holds precisely when every irreducible factor of F occurs\n  with multiplicity at least τ.\n\n4.  Deduce the degree constraint  \n\n    r m ≤ s (m−1)                                 (**)\n\n  and show that (**) is automatic once (†) is satisfied.\n\n5.  Describe (★) in one variable (n=1):  \n  write F(t)=∏_{j}(t−aⱼ)^{eⱼ}.  \n  Then (★) is equivalent to eⱼ ≥ τ for every root aⱼ; there may be several\n  distinct roots, each occurring with multiplicity at least τ.  \n  In particular, when τ>m the only possibility is impossible, hence (★)\n  cannot hold.\n\nThe task is to establish the statements above.",
      "solution": "Throughout let  \n\n  Gᵢ := ∂F/∂xᵢ (i=1,…,n), m:=deg F, F=c·∏_{j=1}^{k}gⱼ^{eⱼ}.  \n\nStep 0.  Extracting individual identities.  \nBecause JFˢ ⊂ (F)ʳ, for every i there is Hᵢ∈R with  \n\n  Gᵢˢ = Fʳ·Hᵢ.                                         (1)\n\nStep 1.  The basic valuation inequality.  \nFix an index j (1≤j≤k) and let vⱼ( · )=v_{gⱼ}( · ) be the gⱼ–adic valuation.\nSince char 𝐊=0, gⱼ and at least one of its partial derivatives are\ncoprime; hence there exists an index i(j) with gⱼ ∤ ∂gⱼ/∂x_{i(j)} and  \n\n  vⱼ(G_{i(j)}) = eⱼ−1.                                 (2)\n\nApply vⱼ to (1) with i=i(j):\n\n  s(eⱼ−1) = vⱼ(G_{i(j)}^{s}) = vⱼ(F^{r})+vⱼ(H_{i(j)})\n            = r eⱼ + vⱼ(H_{i(j)}) ≥ r eⱼ.               (3)\n\nTherefore  \n\n  (s−r)·eⱼ ≥ s  for every j,  so s>r                (4)\n\nand inequality (†) follows.  Setting  \n\n  τ := ⌈s/(s−r)⌉ ≥ 2                                  (5)\n\nwe have eⱼ ≥ τ for all j, finishing Part 1.\n\nStep 2.  The converse.  \nAssume now s>r and eⱼ ≥ τ for every j, equivalently\n\n  s(eⱼ−1) ≥ r eⱼ  for all j.                         (6)\n\nTake any generator P := G_{i₁}⋯G_{i_s} of JFˢ.  \nFor every j,\n\n  vⱼ(P) = vⱼ(G_{i₁})+⋯+vⱼ(G_{i_s})\n          ≥ s(eⱼ−1)             (by (2) and vⱼ≥eⱼ−1)\n          ≥ r eⱼ.                (by (6))\n\nThus vⱼ(P) ≥ vⱼ(F^{r}) for all j, i.e. F^{r} divides P.  As the P generate\nJFˢ we have JFˢ ⊂ (F)ʳ.  This proves Part 2.\n\nStep 3.  Classification.  \nCombining Parts 1 and 2 we obtain\n\n  JFˢ ⊂ (F)ʳ  ⇔  s>r and eⱼ ≥ τ (all j)  \n  ⇔  s>r and (rad F)^{τ} | F,                       (7)\n\ni.e. statement 3.\n\nStep 4.  The degree inequality.  \nFrom (4) we already have s>r.  Summing (†) over j gives  \n\n  (s−r)·m = (s−r)·∑_{j} eⱼ·deg gⱼ\n            ≥ s·∑_{j} deg gⱼ = s·deg(rad F) ≥ s,      (8)\n\nwhence (s−r)·m ≥ s, i.e. r m ≤ s (m−1).  Therefore (**) is a direct\nconsequence of (†).\n\nStep 5.  The one–variable case.  \nFor n=1 we have JF = (F′).  Condition (★) reads (F′)ˢ ⊂ (F)ʳ, i.e.\nFʳ divides (F′)ˢ.  Writing F(t)=∏_{j}(t−aⱼ)^{eⱼ} we get exactly the same\ninequalities s(eⱼ−1) ≥ r eⱼ, hence eⱼ ≥ τ for every root aⱼ.  Several\ndistinct roots may occur, but each with multiplicity at least τ.\nConversely, if this holds then (★) is satisfied, completing the\ndescription.\n\nAll requested statements are now fully proved.",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.370868",
        "was_fixed": false,
        "difficulty_analysis": "1. Higher Dimension & Ideals  \n  • The original statement concerns a single-variable polynomial and one ordinary derivative.  \n  • The enhanced variant moves to n ≥ 2 variables and replaces “divisibility of powers’’ by an **ideal–theoretic containment** involving the whole Jacobian ideal and its s-th power.  \n\n2. Additional Algebraic Structures  \n  • The proof now lives in the unique-factorisation domain 𝐊[x₁,…,xₙ] and uses valuations along irreducible hypersurfaces, primary-decomposition reasoning and degree considerations inside multivariate polynomial rings.  \n\n3. Stronger Conclusion via Finer Arguments  \n  • One must first deduce that **no new irreducible factors** may appear in any partial derivative – a subtle argument that forces every Hᵢ in (1) to be a scalar.  \n  • From there, degree-matching pins down exact numerical relations among r, s and m (equation (7)), something absent from the original exercise.  \n\n4. Interacting Concepts  \n  • Techniques from algebraic geometry (Jacobian ideals and singular loci), commutative algebra (ideal powers, valuations) and elementary algebra (degree counting) all co-operate.  \n  • The case-by-case elimination of multiple or non-linear factors requires repeated, carefully chosen valuations and exploits characteristic 0 in an essential way.\n\nIn short, the enhanced variant is significantly harder because it blends multivariate polynomial theory, ideal-containment arguments and valuation tools, all absent from the original single-variable setting, and forces the solver to navigate several sophisticated layers before the simple “power of a linear form’’ conclusion emerges."
      }
    },
    "original_kernel_variant": {
      "question": "Let 𝐊 be an algebraically closed field of characteristic 0 and put  \n  R = 𝐊[x₁,…,xₙ] \\;(n ≥ 1).  \n\nFor a non–constant polynomial F∈R of total degree m≥1 set  \n\n  JF := (∂F/∂x₁,…,∂F/∂xₙ)  ⊂ R,  JFˢ := (JF)ˢ\n\n(the ordinary s-th power of the Jacobian ideal).  \nFix positive integers r,s and assume  \n\n  (★)  JFˢ ⊂ (F)ʳ.  \n\nDenote by rad F the radical ∏ gᵢ of the principal ideal (F) (product of the\ndistinct irreducible factors of F) and write  \n\n  F = c·g₁^{e₁}⋯g_k^{e_k} (c∈𝐊*, gᵢ irreducible, eᵢ≥1).  \n\nDefine                                                            \n\n  τ := ⌈ s /(s−r) ⌉ (so τ≥2 and s>r).\n\n1.  Prove that s>r and that every multiplicity eᵢ satisfies the sharp\n  inequality                                                     \n\n    (s−r)·eᵢ  ≥  s      (†)  \n\n  equivalently eᵢ ≥ τ for all i.\n\n2.  Conversely, assume s>r and eᵢ ≥ τ for every irreducible divisor gᵢ of F.\n  Show that (★) holds.\n\n3.  Conclude the following equivalence (classification of (★)):  \n\n  JFˢ ⊂ (F)ʳ ⇔ s>r and F is divisible by (rad F)^{τ}.  \n\n  In words: (★) holds precisely when every irreducible factor of F occurs\n  with multiplicity at least τ.\n\n4.  Deduce the degree constraint  \n\n    r m ≤ s (m−1)                                 (**)\n\n  and show that (**) is automatic once (†) is satisfied.\n\n5.  Describe (★) in one variable (n=1):  \n  write F(t)=∏_{j}(t−aⱼ)^{eⱼ}.  \n  Then (★) is equivalent to eⱼ ≥ τ for every root aⱼ; there may be several\n  distinct roots, each occurring with multiplicity at least τ.  \n  In particular, when τ>m the only possibility is impossible, hence (★)\n  cannot hold.\n\nThe task is to establish the statements above.",
      "solution": "Throughout let  \n\n  Gᵢ := ∂F/∂xᵢ (i=1,…,n), m:=deg F, F=c·∏_{j=1}^{k}gⱼ^{eⱼ}.  \n\nStep 0.  Extracting individual identities.  \nBecause JFˢ ⊂ (F)ʳ, for every i there is Hᵢ∈R with  \n\n  Gᵢˢ = Fʳ·Hᵢ.                                         (1)\n\nStep 1.  The basic valuation inequality.  \nFix an index j (1≤j≤k) and let vⱼ( · )=v_{gⱼ}( · ) be the gⱼ–adic valuation.\nSince char 𝐊=0, gⱼ and at least one of its partial derivatives are\ncoprime; hence there exists an index i(j) with gⱼ ∤ ∂gⱼ/∂x_{i(j)} and  \n\n  vⱼ(G_{i(j)}) = eⱼ−1.                                 (2)\n\nApply vⱼ to (1) with i=i(j):\n\n  s(eⱼ−1) = vⱼ(G_{i(j)}^{s}) = vⱼ(F^{r})+vⱼ(H_{i(j)})\n            = r eⱼ + vⱼ(H_{i(j)}) ≥ r eⱼ.               (3)\n\nTherefore  \n\n  (s−r)·eⱼ ≥ s  for every j,  so s>r                (4)\n\nand inequality (†) follows.  Setting  \n\n  τ := ⌈s/(s−r)⌉ ≥ 2                                  (5)\n\nwe have eⱼ ≥ τ for all j, finishing Part 1.\n\nStep 2.  The converse.  \nAssume now s>r and eⱼ ≥ τ for every j, equivalently\n\n  s(eⱼ−1) ≥ r eⱼ  for all j.                         (6)\n\nTake any generator P := G_{i₁}⋯G_{i_s} of JFˢ.  \nFor every j,\n\n  vⱼ(P) = vⱼ(G_{i₁})+⋯+vⱼ(G_{i_s})\n          ≥ s(eⱼ−1)             (by (2) and vⱼ≥eⱼ−1)\n          ≥ r eⱼ.                (by (6))\n\nThus vⱼ(P) ≥ vⱼ(F^{r}) for all j, i.e. F^{r} divides P.  As the P generate\nJFˢ we have JFˢ ⊂ (F)ʳ.  This proves Part 2.\n\nStep 3.  Classification.  \nCombining Parts 1 and 2 we obtain\n\n  JFˢ ⊂ (F)ʳ  ⇔  s>r and eⱼ ≥ τ (all j)  \n  ⇔  s>r and (rad F)^{τ} | F,                       (7)\n\ni.e. statement 3.\n\nStep 4.  The degree inequality.  \nFrom (4) we already have s>r.  Summing (†) over j gives  \n\n  (s−r)·m = (s−r)·∑_{j} eⱼ·deg gⱼ\n            ≥ s·∑_{j} deg gⱼ = s·deg(rad F) ≥ s,      (8)\n\nwhence (s−r)·m ≥ s, i.e. r m ≤ s (m−1).  Therefore (**) is a direct\nconsequence of (†).\n\nStep 5.  The one–variable case.  \nFor n=1 we have JF = (F′).  Condition (★) reads (F′)ˢ ⊂ (F)ʳ, i.e.\nFʳ divides (F′)ˢ.  Writing F(t)=∏_{j}(t−aⱼ)^{eⱼ} we get exactly the same\ninequalities s(eⱼ−1) ≥ r eⱼ, hence eⱼ ≥ τ for every root aⱼ.  Several\ndistinct roots may occur, but each with multiplicity at least τ.\nConversely, if this holds then (★) is satisfied, completing the\ndescription.\n\nAll requested statements are now fully proved.",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.320711",
        "was_fixed": false,
        "difficulty_analysis": "1. Higher Dimension & Ideals  \n  • The original statement concerns a single-variable polynomial and one ordinary derivative.  \n  • The enhanced variant moves to n ≥ 2 variables and replaces “divisibility of powers’’ by an **ideal–theoretic containment** involving the whole Jacobian ideal and its s-th power.  \n\n2. Additional Algebraic Structures  \n  • The proof now lives in the unique-factorisation domain 𝐊[x₁,…,xₙ] and uses valuations along irreducible hypersurfaces, primary-decomposition reasoning and degree considerations inside multivariate polynomial rings.  \n\n3. Stronger Conclusion via Finer Arguments  \n  • One must first deduce that **no new irreducible factors** may appear in any partial derivative – a subtle argument that forces every Hᵢ in (1) to be a scalar.  \n  • From there, degree-matching pins down exact numerical relations among r, s and m (equation (7)), something absent from the original exercise.  \n\n4. Interacting Concepts  \n  • Techniques from algebraic geometry (Jacobian ideals and singular loci), commutative algebra (ideal powers, valuations) and elementary algebra (degree counting) all co-operate.  \n  • The case-by-case elimination of multiple or non-linear factors requires repeated, carefully chosen valuations and exploits characteristic 0 in an essential way.\n\nIn short, the enhanced variant is significantly harder because it blends multivariate polynomial theory, ideal-containment arguments and valuation tools, all absent from the original single-variable setting, and forces the solver to navigate several sophisticated layers before the simple “power of a linear form’’ conclusion emerges."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}