PηooヅIn a subsequent proof，we uβe the fo11owing resu1t due to Kojima−
Miyanishi［101［Theorem1．21．
lLemma4．2．2．〃B二ψ2，＿， 。，μ2，＿，〃αη〃θ伽θδ∈LNDん（B）的
δ（η）＝0αηdδ（ω＝ づ七十1∫orθαcん乞．丁加ηKerδ乞8αポαZgθうm geηem加∂
的・。，＿，・肌αη畑士十1μr・！十1μ｛（2≦1，ゴ≦η，1≠ゴ）．
For each monom1a1m＝ 1α1 叫α几V161 ％斗1bれ斗1，de丘ne
τ（m）一
u亡羊11・・［之辛11一（1・… 一）・
Let∫1，n＋1＝ ！V、十1＿（ 2、．、 肌）〜1and1et∫｛，ゴ＝軌七十1坊＿ ！斗1晩for each pair（乞，ゴ）with1≦乞，プ≦れand−4≠ゴ、It is immediate that a11of∫1，η十1，
吻and篶，ゴbe1ong toλ． Letλノbe theん＿suba1gebra generated by∫1，肌十1，
均（1≦乞≦η）and力，ゴ（1≦乞，ゴ≦η，乞≠ゴ）．In the same fashion as in the proof of Theorem4．1．3，it su田。es to show that七here exists∫∈〃such that
∫1，肌十14＿∫is of the form
1勾九十1乏十 14−1（terms of工。wer degree in％十1）．
We suppose that we obtain an e1ement in〃。fthe form
G、：・。㌦、。。七十9。一。μ、。。4−1＋9仁。μ、。。ゼ 2＋・十9、μ肌。。「十・十9。
with g乞∈ん［ 1，．．、， 、，眈，．．．，gれ］and g4＿1，．．．，g、十1divisib1e by 14−1，and
we use the descending induction0H，We show that G，is modi丘ed by an e1ement ofλ so that a new g，is divisib1e by 1H without changing the terms gH，＿，g、十1．Furthermore，we suppose the fo11owing cond−itions are satis丘ed．
（1）F・・0≦1≦グ11ifw・w・it・g1一Σゴ市・州1，lwith〜∈
ψ・，．一．，・、，μ・，＿，μ、1；th・nゴ十ト（亡十1）9台，ゴ＝（亡十1）4．
（2）We have仙，o＝…＝仏、乞一1＝0for0≦づ≦4−1，i．e．，for each
1ゴμ1〜伽，ゴapPearing in g乞，we haveゴ≧乞．
（3）For each monomia1m：α1ゴ物α2＿ 肌αれμ1b1．．．％b・％十1｛in
μ、斗1㌦1ゴμ1舳伽，ゴwithゴ≦4＿2，each ofα2，＿，αれis congruent toτ（4＿ゴ）
mod−u1o亡十1and if we writeゴ 三4一ゴ（mod一む十1）with0≦ゴ ≦亡，
we have
4．2．A GENERA∬ZAT工ON O−F THEOREM 4．ユ．3 65
（i） （t＋1）τ（η7）≧4一ゴ十2（老一2）Provided．ク ＝0，
（ii）（t＋1）τ（m）≧4一ゴ十2ゴノー6p・・vid・dプ ＞0．
The condition（3）imp1ies that for each monomia1m in％十1毒 1ゴV1q4，仏，ゴwith ゴ≦4−2，we haveτ（m）≧0．Indeed一，it is c1ear that4一ク十2（t−2）≧0．
Ifグ＝1，then since4一ゴ≧2andむ十1≧3，we have4川ゴ≧4and hence
4一ゴ十2ゴ 一6＝4一ゴー4≧0．Ifア≧2，then we have4一ゴ十2ゴ 一6≧2＋2．2＿6＝0．
In the same way as in Theorem4．1．3，we suppose by a doub1e ind−uction
thatん、，o＝…＝ん、，ρ＿1：0andん、，ρ≠0with r≦p≦4−2，and d−enote G，by G、，、．The beginning po1ynomia1for induction is G4＿2，4＿2＝∫1，、十14，for whi・hgF（一1）4一｛（1）・・乞（（・。…・、）W一｛，ん｛，F（一1）4一づ（1）（・・…・肌）亡（4一｛），
伽，ゴ＝0（乞≠ゴ）and砺，づ：4＿乞for O≦4，ゴ≦4＿1．One can check easi1y that the above conditions（1）and（2）are satis丘ed for G4−2，4−2＝∫1，九十14．We give
・・1y・p…f・fth・…diti・・（3）f・・G。一。，。一。＝∫。、肌。。4，F・・…hl≦ト2，
take乞 ≡4一乞with0≦乞 ≦オ．The exponents of 2ゾ．．，叫in（ 2…叫）壬（4−4）
are aII equaI toψ＿乞）moduユ。亡十1．If〆＝0，we have
（九十1）τ（凶几。。4（（・。…π、）士V。）4−4）＝（上1）（4−1）亡一（亡十1）（ト1）
≧2左（ト4）ト（之十1）（ト1）＝（t−1）（グづ）＝（4一乞）十（亡一2）（4一つ）
≧4一十2（亡一・2），
where we use the cond−itionη≧4，乏≧2and乞≦4−2．If乞 ＞0，we have
（亡十1）巾。㌦。。｛（（・。…・、）士g1）4 乞）
一（一・）（・一… ）1・（一・）（1・1）［亡羊11一（1・・）（H）
・・（・一…∫）1・・（1・1）［⑫・一亡羊11一（1・1）（H）
＝（ト1）（亡一1）一助 十2（之十1）（1 一1）＝（4一乞）（ト1）一2之十21L2
＝（4一乞）十（4−1）（之一2）一2亡十21L2
≧トづ十2（之一2）一2亡十2乞 一2＝4一つ十2乞 一6，
where we use the conditionη≧4，乞≧2andゼ≦4＿2．
We exp1ain the process of improving g、．By the same reasoning as in
Theorem4．1．3，we haveδ（ん。，ρ）＝0．Lemma4．2，2imp1ies tha七ん、，ρis a sum
66
CHAPTER4．工NF工N工TELγMANγGENERATORS
of po1ynomia1s of the form
…d2・小■九ノ毛・1
｛，ゴ∈｛2，＿、η｝
with c∈たand non−negative integersφ，ら，ゴーNote that a11of d2，＿，dηare congruent toψ＿ρ）modu1o亡十1，In fact，since the contributions ofthe力，ゴ to the exponent d2，．．．，dηare a mu1tip1e of古十1，the remark fo11ows from the condition（3）．Now wb choose any one of the above po1ynomia1s and1et
H一π、，ゴ、｛。，，、｝∫1，！｛・ゴT・k・p∫≡トρ（m・dl＋1）with0≦ρノ≦1・Th・・
for each monomia1m in％十1「 1ρμ1伽 2d2＿ ノ皿∬，we have in view of（i）
・nd（ii）・f（3），
（亡十1）τ（μ。・・…。d2…・几d・）：（亡十1）τ（m）
・にlll）：ll；｝1続，
where mu1tip1yingμ1舳吻d2．．．エ肌dηby any ｛f＋1坊（乞≠1），蝪十10r 1does not change the va1ue ofτ．As we remark above，we have
（t＋1）τ（μ。伽…。d・…・れdれ）＝（1＋1）τ（m）≧0．
Hence there exists an eIement F∈〃。f the form
F一・バ「九，几。。「危，。q・…∫肌，。伽・。d・■（士十1）・・…・れ∂r（士十1）％
：・・。ρﾊ。・・，・μれ。。「・。d・…・肌d一
十（terms of degree＞ρin 1）十（terms of degree＜r in蝪十1），
where g2＋＿十％：g。，ρ．We can prove that G、，p＿F∬sa七is丘es the same conditions as G、，ρd−oes except for the conditionん、，ρ≠0but the number of nonzero terms inん、，ρgets sma11er．We prove this be1ow，By repeating this process inite1y many times，we obtain a new G，satisfying the condition ん、、，ρ＝0．Fur七her，continuing this process丘ni七e1y many times，we obtain a mod．i丘ed−G，satisfying the conditionん、，o＝… ＝ん、，4＿2＝0，i．e．，g，is divisib1e by 1ゼー1．Hence by induction on r，we comp1ete a proof．
Now we show that G。，ρ一FH satis丘es七he same conditions as G、，ρd−oes but the number of nonzero monomia1terms inん、，ρbecomes1ess．We have on1y to show that each monomia1in F satis丘es七he cond−itions（1）一（3）since