Journal Article

Vojta’s refinement of the subspace theorem

Schmidt, Wolfgang M.

1993

Overview

Vojta’s refinement of the Subspace Theorem says that given linearly independent linear forms L1,…,Ln{L_1}, \\ldots , {L_n} in nn variables with algebraic coefficients, there is a finite union UU of proper subspaces of Qn{\\mathbb {Q}^n}, such that for any ε>0\\varepsilon > 0 the points x__∈Zn∖{0__}\\underline {\\underline x} \\in {\\mathbb {Z}^n}\\backslash \\{ \\underline {\\underline 0} \\} with (1) |L1(x__)⋯Ln(x__)|>|x__|−ε|{L_1}(\\underline {\\underline x} ) \\cdots {L_n}(\\underline {\\underline x} )|\\; > \\;|\\underline {\\underline x} {|^{ - \\varepsilon }} lie in UU, with finitely many exceptions which will depend on ε\\varepsilon . Put differently, if X(ε)X(\\varepsilon ) is the set of solutions of (1), if X¯(ε)\\bar X(\\varepsilon ) is its closure in the subspace topology (whose closed sets are finite unions of subspaces) and if X¯′(ε)\\bar X\\prime (\\varepsilon ) consists of components of dimension >1> 1 , then X¯′(ε)⊂U\\bar X\\prime (\\varepsilon ) \\subset U . In the present paper it is shown that X¯′(ε)\\bar X\\prime (\\varepsilon ) is in fact constant when ε\\varepsilon lies outside a simply described finite set of rational numbers. More generally, let kk be an algebraic number field and SS finite set of absolute values of kk containing the archimedean ones. For υ∈S\\upsilon \\in S let L1υ,…,LmυL_1^\\upsilon , \\ldots ,L_m^\\upsilon be linear forms with coefficients in kk, and for x__∈Kn∖{0__}\\underline {\\underline x} \\in {K^n}\\backslash \\{ \\underline {\\underline 0} \\} with height Hk(x__)>1{H_k}(\\underline {\\underline x} ) > 1 define aυi(x__){a_{\\upsilon i}}(\\underline {\\underline x} ) by |Liυ(x__)|υ/|x__|υ=Hk(x__)−aυi(x__)/dυ|L_i^\\upsilon (\\underline {\\underline x} )|_\\upsilon /|\\underline {\\underline x} |_\\upsilon = {H_k}{(\\underline {\\underline x} )^{ - {a_{\\upsilon i}}(\\underline {\\underline x} )/{d_\\upsilon }}} where the dυ{d_\\upsilon } are the local degrees. The approximation set AA consists of tuples a__={aυi}(υ∈S,1≦i≦m)\\underline {\\underline a} = \\{ {a_{\\upsilon i}}\\} \\;(\\upsilon \\in S,1 \\leqq i \\leqq m) such that for every neighborhood OO of a__\\underline {\\underline a} the points x__\\underline {\\underline x} with {avi{x__)}∈O\\{ {a_{{v_i}}}\\{ \\underline {\\underline x} )\\} \\in O are dense in the subspace topology. Then AA is a polyhedron whose vertices are rational points.

• Publisher: American Mathematical Society
• Subject: Algebra / Approximation / Coefficients / Exact sciences and technology / Integers / Mathematical theorems / Mathematics / Number theory / Polyhedrons / Research article / Sciences and techniques of general use / Topological theorems / Topology / Variable coefficients / Vertices