On two extremum properties of polynomials
On two extremum properties of polynomials
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关于多项式的两个极值性质
DOI：
10.1215/ijm/1255645104
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发表时间：
1963
影响因子：
0.6
通讯作者：
K. Mahler
中科院分区：
文献类型：
--
作者：
K. Mahler
本文研究了具有实系数或复系数的任意多项式f(x) aox - kal x + + am的测度M(f)的某些极值性质。在丢芬图近似理论中，特别是在超越数理论中，通常用它的高度H(f) max（a0], laX 1，“'”，am I）或它的长度I (f)]a0[+ lal / ..]来表示这种u多项式的大小。——/ laml。这里的长度具有伪值的优点：(1)L(fg) -<_ L(f)L(g)， L(f =V g) <= L(f) -}-L(g)。还有另一个系数函数，由于其简单的乘法性质，值得考虑。这是f(x)的M(f)它由M(f) 0定义如果f(x) 0, exp logIf(e 'it) Id否则。
Introduction This paper is concerned with certain extremum properties of the measure M(f) of an arbitrary polynomial f(x) aox -k al x + + am with real or complex coefficients. In the theory of Diophantine approximations, and in particular in that of transcendental numbers, it hs been customary to express the size of such u polynomial either by its height H(f) max( a0 ], laX l, "’", am I), or by its length i(f) ]a0[ + lal / ..-/ laml. Here the length has the advantage of being pseudo-valuation: (1) L(fg) -<_ L(f)L(g), L(f =V g) <= L(f) -}-L(g). There is still another function of the coefficients which is worth considering on account of its simple multiplictive property. This is the measure M(f) of f(x) which is defined by M(f) 0 if f(x) O, exp logIf(e’it) Id otherwise.