HIRATA-KOHNO, NORIKO

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TitleAN APPLICATION OF QUANTITATIVE SUBSPACETHEOREM(Analytic Number Theory)Author(s)HIRATA-KOHNO, NORIKOCitation数理解析研究所講究録 (1994), 886: 81-87Issue Date1994-09URL http://hdl.handle.net/2433/84312RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityAN APPLICATION OF QUANTITATIVE SUBSPACE THEOREMNORIKO HIRATA-KOHNO $( \ovalbox{\tt\small REJECT} ut ffi_{\backslash }\sim+)$Department of Mathematics, College of Science and Technology,Nihon University, Suruga-Dai, Kanda, Chiyoda) Tokyo 101, Japan$\Gamma$ aX $\neq\sum\Sigma)$1. IntroductionWe give here one kind of generalization of the Schmidt-Schlickewei quantitative subspacetheorem, using the argument of M. Ru and P. -M. Wong. We consider linear forms notonly of the same number as variables but also of a larger number than the number ofvariables.Let $K$ be a normal extention of $Q$ of degree $d$ over Q. Let $M(K)$ be the set of non-equivalent places of $K$ : for $v\in M(K)$ , denote by . $|_{v}$ the corresponding absolute valuenormalized such that for $a\in Q,$ $|a|_{v}=|a|$ if $v$ lies above the archimedean prime of $Q$ ,and $|p|_{v}=1/p$ if $v$ lies above the rational prime $p$ . Let $M_{\infty}(K)$ be the set of archimedeanplaces of $K$ .We consider $S\subset M(K)$ (not necessarily containing $M_{\infty}(K)$), of cardinality $s<\infty$ . Let$K_{v}$ be the completion of $K$ with repect to . $|_{v}$ . Put $d_{v}=[K_{v} : Q_{v}]$ for the local degree.For $a\in K$ , we put $\Vert a\Vert=|a|\lrcorner d^{L}d$ .For $a=$ $(a_{0}, \cdots , a_{n})\in K^{n+1},$ $v\in M(K)$ , write$|$ a $|_{v}=(|a_{0}|_{v}^{2}+\cdots+|a_{n}|_{v}^{2})^{\frac{1}{2}}$if $v$ is archmedean, anda $|_{v}=_{0} \max_{\leq i\leq n}|a_{i}|_{v}$if $v$ is nonarchmedean. Let us denote $\Vert$ a $\Vert_{v}=|$ a $|^{\frac{d_{v}}{v^{d}}}$ . We define the height of a by$H( a)=\prod_{v\in M(K)}\Vert$a $\Vert_{v}$and write $h(a)=\log H(a)$ .It is well-known that the definition of $H(a)$ is independent of a choice of field where a lies,and also that $H(ca)=H(a)$ for $c\in K-\{0\}$ . Given a linear form $L(x)=a_{0}x_{0}+\cdots+a_{n}x_{n}$Typeset by $\eta$ STffi886 1994 81-87 81with coefficients $a_{0},$ $\cdots a_{n},$ $\in K$ not all zero, we put $H(L)=H(a)$ for $a=(a_{0}\cdots a_{n})$ asthe height of $L$ . For $v\in M(K)$ we denote $\Vert L\Vert_{v}=\Vert$ a $\Vert_{v}$ .2. Quantitative subspace theoremA higher dimensional case of Roth’s theorem, that we call the subspace theorem, isestablished by W. M. Schmidt for archimedean places. A quantitative version is alsoderived by himself, and extended by H. P. Schlickewei to nonarchimedean places. See forthe historical survey in [Schl] [Schm 1] [Schm 2]. We apply here the theorem of Schlickeweiin [Schl] which is stated as follows.Theorem 2.1 (Schlickewei).Let $K,$ $S$ as above. Suppose that for each $v\in S$ we are $gii^{\gamma}enn+1$ lineaxly independentlineax forms $L_{1}^{(v)},$ $\cdots,$ $L_{n+1}^{(v)}$ in $n+1$ variables with coefficients in K. Let $0<\delta<1$ .Consider the inequfflty$\prod_{v\in S}\prod_{i=1}^{n+1}\frac{||L_{i}^{(v)}(x)\Vert_{v}}{\Vert L_{i}^{(v)}||_{v}\Vert x\Vert_{v}}<H(x)^{-n-1-\delta}$ .Then $tAere$ exists proper $su$bspaces $S_{1},$ $\cdots,$ $S_{t_{1}}$ of $K^{n+1}wit\Lambda$$t_{1}=[(8sd)^{2^{34(n+1)d}s^{6}\delta^{-2}}]$$sucA$ that every solution $x\in K^{n+1}$ lies in$\bigcup_{i=1}^{t_{1}}S_{i}\cup D$$wAere$$D= \{x\in K^{n+1} ; H(x)<\max((n+1)!\frac{9}{\delta}, H(L_{i}^{(v)})\frac{9d(n+1)s}{\delta}(v\in S, i=1, \cdots, n+1))\}$ .3. PreliminariesWe recall here the definition of subgeneral position and Nochka weight following [R-W].Let $1\leq k\leq n<q$ be rational integers. Consider nonzero distinct $q$ linear forms in $k+1$variables with coefficients in $K$ . Fbr each linear form $L_{i}(x)=a_{i0}x_{0}+\cdots+a_{ik^{X}k}$ , put$a_{i}=a(L_{i})=(a_{i0}, \cdots, a_{ik})\in K^{k+1}$ $(1 \leq i\leq q)$ . The linear forms $L_{1},$ $\cdots,$ $L_{q}$ are calledin n-subgeneral position if any distinct $n+1$ elements of the set $\{a_{1}, \cdots, a_{q}\}$ span $K^{k+1}$ .We see that n-subgeneral position is equivalent to general position when $n=k$ .Now we define Nochka weight (cf [R-W]).Let $1\leq k\leq n<q$ be rational integers and $L_{1},$ $\cdots,$ $L_{q}$ be linear forms in $k+1$ variableswith coefficients in $K$ , supposed to be in n-subgeneral position. We denote the dimension of82the linear span over $K$ of a subset $B\subset A$ $:=\{a_{1}, \cdots , a_{q}\}$ by $d(B)$ . Put $P(B)=(\# B, d(B))$which is regarded as a point in $R^{2}$ . For two points $P_{1}=(x_{1}, y_{1}),$ $P_{2}=(x_{2}, y_{2})$ in $R^{2}$ with$x_{1}\neq x_{2}.$ , we write $\sigma(P_{1}, P_{2})=L1^{-}\ovalbox{\tt\small REJECT} 2x_{1}-x_{2}$ . Proposition 2.1 in [R-W] (under some corrections)allows us to show that there exists a sequence of subsets$A=B_{s+1}\supset B_{s}\supset B_{s-1}\cdots\supset B_{1}\supset B_{0}=\emptyset$where the sequence of numbers $\sigma(P(B_{i+1}), P(B_{i}))$ $(0\leq i\leq s)$ is uniquely determined.If an element $a\in A$ lies in $B_{i+1}-B_{i}(0\leq i\leq s)$ , we put $\omega(a)=\sigma(P(B_{i+1}), P(B_{i}))$ ,which is called Nochka weight. For simplicity, we write $\sigma(P(B_{i+1}), P(B_{i}))=\sigma_{i}$ . Severalproperties of Nochka weight are presented in [R-W].4. ResultsFor simplicity, we restrict here $K\subset R$ and consider $S=\{\infty\}$ ; one archmedean placeof $K$ defined by $|x|_{\infty}= \max(x, -x)$ . Put $|x|_{\infty}=|x$ .For $1\leq k\leq n<q$ , consider lineax forms $L_{1},$ $\cdots,$ $L_{q}$ in $k+1$ variables with coefficients in$K$ , supposed to be in n-subgeneral position. Write $a_{i}=(a_{i0}, \cdots, a_{ik})$ a coefficient vectorof $L_{i}$ respectively $(1 \leq i\leq q)$ . Then for all $x\in R^{k+1}$ we claim$\#\{i;\frac{\Vert L_{i}(x)\Vert}{\Vert L_{i}||\Vert x\Vert}<c_{0}\}\leq n$with$c_{0}= \frac{1}{2}m\dot{j}_{a}na:j(1-\frac{|(a_{i},a_{j})|}{|a_{i}||a_{j}|})^{\frac{d}{2}}-$where $(a_{i}, a_{j})=a_{0i}a_{0j}+\cdots+a_{ki}a_{kj}$ .Using Theorem 2.1, we get the following quantitative statement of Theorem 3.3 of [R-$W]$ .Theorem 4.1.Let $K,$ $S$ as above.Let $1\leq k\leq n<q$ be rational integers and $L_{1},$ $\cdots,$ $L_{q}$ be linear forms in $k+1$ variables$witI_{J}$ coefficients in $K$ , supposed to be in n-subgeneral position. Let $\omega i=\omega(a_{i})$ be theassociated Nochka weight $w^{r}ithL_{i}$ $(1\leq i\leq q)$ . Let $0<\delta<1$ . Consider the inequality$\sum\omega_{i}\log q(\frac{\Vert L_{i}||\Vert x\Vert}{\Vert L_{i}(x)\Vert})>(k+1+\delta)\log|x$ .$i=1$Then there exists proper $su$bspaces $S_{1},$ $\cdots,$ $S_{t_{2}}$ of $K^{k+1}$ with$t_{2}=[32d^{2^{34(k+1)d}\delta^{-2}}]$83$;$$| x|<\exp(\frac{2c_{1}}{\delta})\}$ ,such that $ei\prime ery$ solution $x\in Z^{k+1}wit\Lambda L_{i}(x)\neq 0$ for all $1\leq i\leq q$ lies in$\cup S_{i}\cup D_{1}\cup D_{2}t_{2}$$i=1$ivbere$D_{1}=\{x\in Z^{k+1}$$D_{2}=\{$ $x\in Z^{k+1}$ ; $H(x)< \max((k+1)!^{\frac{18}{\delta}}, H(L_{i})\frac{18d(k+1)}{\delta})\}$and$c_{1}= \frac{(q-n)(k+1)}{n+1}\log\frac{1}{c_{0}}$ .Outline of the proof of Theorem 4.1We follow the argument of Ru-Wong. Take $c_{0}$ as above. For $1\leq i\leq q$ , put $E_{i}(x)=$$\frac{||L;||||_{J}\epsilon||}{||L_{i}(x)||}$ . Then we have $\# I(x)\leq n$ where $I( x)=\{i;\log E_{i}(x)\geq\log\frac{1}{c_{0}}\}$ . Lemma 3.1 of[R-W] implies that there exists a set $J(x)$ of cardinality $k+1$ such that $\{a_{i};i\in J(x)\}$ arelinearly independent and$\prod_{i\in I(x)}E_{i}(x)^{\omega;}\leq\prod_{i\in J(x)}E_{i}(x)$with $\omega_{i}=\omega(a_{i})$ for $L_{i}(x)=(a_{i}, x)$ . Therefore$\prod_{i\in J(x)}\max_{i\in I}$where $I$ runs over the family of all subsets of $\{$ 1, $\cdots,$ $q\}$ with $\# I=k+1$ and $\{a_{i};i\in I\}$linearly independent. Using the property $\omega_{i}\leq\frac{k+1}{n+1}$ of Nochka weight, we obtain$\sum_{i=1}^{q}\omega_{i}\log E_{i}(x)=\sum_{i\in I(x)}\omega_{i}\log E_{i}(x)+c_{1}\leq\max_{I}\sum_{i\in I(x)}\log E_{i}(x)+c_{1}$with $c_{1}= \frac{(q-n)(k+1)}{n+1}\log\frac{1}{c_{0}}$ . Thus the solutions $x\in Z^{k+1}$ outside of $L_{1},$ $\cdots,$ $L_{q}$ of theinequality$\sum_{i=1}^{q}\omega_{i}\log E_{i}(x)>(k+.1+\delta)\log|x|$are contained in the solutions $x\in Z^{k+1}$ outside of $L_{1},$ $\cdots,$ $L_{q}$ of the inequality(4.2) $\max\sum_{i\in I}^{q}\log E_{i}(x)>(k+1+\delta)\log|x|-c_{1}$ .84Then the solutions of (4.2) are contained in the union of the set $D_{1}$ and the set of thesolutions of(4.3) $\max\sum_{i\in I}^{q}\log E_{i}(x)>(k+1+\frac{\delta}{2})\log|x|$ ,because the solutions of (4.2) with $\log|x|\geq\frac{2c_{1}}{\delta}$ satisfies$(k+1+ \delta)\log|x|-c_{1}\geq(k+1+\frac{\delta}{2})\log|x$ .Now we apply Theorem 2.1 for $k+1$ variables to solve (4.3) which establishes our statement.Applying this theorem, we obtain a quantitative statement of Theorem 3.5 of [R-W] asfollows.$;$ $| x|<\exp(\frac{4(2n+1)c_{1}}{\delta})\}$ ,Theorem 4.4.Let $1\leq k\leq n<q$ ivith $q>2n-k+1$ be rational integers and $L_{1},$ $\cdots,$ $L_{q}$ be lineaxforms in $k+1$ variables witb coefficients in $K$ , in n-subgeneral position.Let $0<\delta<1$ . Consider $tl1e$ inequality$\sum_{i=1}^{q}\log$ $\frac{\Vert L_{i}||\Vert x\Vert}{\Vert L_{i}(x)\Vert}$ $>(2n-k+1+\delta)\log|x$ .Then ihere exists proper. subspaces $S_{1},$ $\cdots$ , $S_{t_{3}}$ of $K^{k+1}$ witk$t_{3}=[128d^{2^{34(k+1)d}(2n+1)^{2}\delta^{-2}}]$$suc\ln$ ihat every solution $x\in Z^{k+1}$ witb $L_{i}(x)\neq 0$ for all $1\leq i\leq q$ lies in$\cup S_{i}\cup D_{3}\cup D_{4}t_{3}$$i=1$where$D_{3}=\{x\in Z^{k+1}$$D_{4}=\{x\in Z^{k+1}$ ; $H(x)< \max((k+1)!\frac{36(2n+1)}{\delta}$ ,$H(L_{i}) \frac{36d(2n+1)(k+1)}{\delta}(i=1, \cdots, q))\}$with $c_{1}$ in Theorem 4.1.85Outline of the proof of Theorem 4.4Put $\theta=\frac{1}{\sigma_{\sim}Q}$ . Then $\theta=\frac{q-2n+k-1}{\omega_{1}+\cdots+\omega_{q}-(k+1)}$ and $\frac{n+1}{k+1}\leq\theta\leq\frac{2n-k+1}{k+1}\leq 2n+1$ byTheorem 4.3 (1) (3) of [R-W]. For all $v\in M(K)$ and for $x$ outside.of zeroes of $L_{i}$ , weclaim that$\log\frac{\Vert L_{i}||_{v}\Vert x\Vert_{v}}{||L_{i}(x)||_{v}}>0$which derives$\log E_{i}(x)\leq\sum_{v\in M(K)}\log\frac{\Vert L_{i}||_{v}\Vert x\Vert_{v}}{||L_{i}(x)\Vert_{v}}=h(a_{i})+h(x)$,because we see $v\in M(K)\log\Vert L_{i}(x)\Vert_{v}=0$ by the product formula. For $\epsilon_{1}>0$ , we getthat a point $x$ either is contained in $D_{5}$ or satisfies $\log E_{i}(x)\leq(1+\epsilon_{1})h(x)$where$D_{5}=l^{x}$ ; $h( x)<\frac{\max_{1\leq i\leq q}h(a_{i})}{\epsilon_{1}}]$ .Therefore for $x\not\in D_{\zeta}$ , we have$\sum_{1\leq i\leq q}\log E_{i}(x)$$= \sum_{1\leq i\leq q}(1-\theta\omega_{i})\log E_{i}(x)+\theta\sum_{1\leq i\leq q}\omega_{i}\log E_{i}(x)$$\leq(1+\epsilon_{1})h(x)\sum_{1\leq i\leq q}(1-\theta\omega_{i})+\theta\sum_{1\leq i\leq q}\omega_{i}\log E_{i}(x)$.Consider $x\in Z^{k+1}$ with $L_{i}(x)\neq 0$ . By theorem 4.1, the inequality$\sum_{1\leq i\leq q}\log E_{i}(x)\leq(1+\epsilon_{1})h(x)(q-\theta\sum_{1\leq i\leq q}\omega_{i})$$+\theta(k+1+\delta_{1})\log|x|$holds for all$x\not\in\bigcup_{i=1}^{t_{2}}S_{i}\cup D_{1}\cup D_{2}\cup D_{5}$ ,with $\delta=\delta_{1}$ in theorem 4.1. Since we have $h(x)\leq\log|x|$ for $x\in Z^{k+1}$ , using the propertyof $\theta$ mentioned above, $x$ satisfies86$\sum_{1\leq-\backslash \leq q}\log E_{i}(x)\leq(1+\epsilon_{1})(2n-k+1-\theta(k+1))\log|x|$$+\theta(k+1+\delta_{1})\log|x|$$=(2n-k+1+\delta_{1}\theta+\epsilon_{1}(2n-k+1-\theta(k+1)))\log|x|$ .For any $0<\delta<1$ , take $\delta_{1}=\epsilon_{1}=\frac{\delta}{2(2n+1)}$ . Then $\epsilon_{1}\leq\frac{\delta}{2(2n-k+1-\theta(k+1))}$ if $2n-k+$$1-\theta(k+1)\neq 0$ , and otherwise we have $\epsilon_{1}(2n-k+1-\theta(k+1))=0$ . This implies$\delta\geq\delta_{1}\theta+\epsilon_{1}(2n-k+1-\theta(k+1))$ which shows that the solutions $x\in Z^{k+1}$ with $L_{i}(x)\neq 0$for all $1\leq i\leq q$ of the inequality in the statement of the Theorem lie in the desired region.REFERENCES[R-W] M. Ru- P. -M. Wong, Integral points of $P^{n}$ –2n $+1$ hyperplanes in general position, Invent.math. 106 (1991)) 195-216.[Schl] H. P. Schlickewei, The quantitative Subspace Theorem for number fields, Compositio Math. 82(1992), 245-273.. [Schm 1] W. M. Schmidt, The subspace theorem in diophantine approximations, Compositio Math. 69(1989) ) 121-173.[Schm 2] W. M. Schmidt, Diophantine Approximations and Diophantine Equations, Lecture Notes inMath., Springer 1467 (1991).87

AN APPLICATION OF QUANTITATIVE SUBSPACE THEOREM(Analytic Number Theory)

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AN APPLICATION OF QUANTITATIVE SUBSPACE THEOREM(Analytic Number Theory)

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