关于多项式周期轨道的乘子的计算(英文)

ON COMPUTATION FOR THE MULTIPLIERS OFPERIODIC ORBITS OF POLYNOMIAL MAPS

设Ｋ为有理数域Ｑ的有限扩张，ｆ是系数在Ｋ中的多项式．令Ωｋ＝｛αｋ，ｆ（αｋ），…，ｆｎ－１（αｋ）｝，ｋ＝１，２，…，ｍ，用Ｈｎ，ｆ表示以ｆ的周期点（周期为ｎ）为根的多项式．如果Ｈｎ，ｆ在Ｋ上不可约，且所有轨道Ωｋ（ｋ＝１，２，…，ｍ）的乘子互不相同，则Ｖａｖａｌｄｉ和Ｈａｔｊｉｓｐｙｒｏｓ给出了一个计算轨道Ωｋ（ｋ＝１，２，…，ｍ）的乘子而无需计算这些轨道本身的算法，在本文中我们证明了Ｖａｖａｌｄｉ和Ｈａｔｊｉｓｐｙｒｏｓ所给的条件是多余的，为此我们证明了对于给定的多项式ｆ＝ｂｋｘｋ＋…＋ｂ１ｘ＋ｂ０，存在一个次数ｋ为的多项式序列ｆｌ＝ｂｋ（ｌ）ｘｋ＋…＋ｂ１（ｌ）ｘ＋ｂ０（ｌ）（ｉ∈Ｎ）使得ｂｊ（ｌ）→ｈｊ（ｌ→∞），ｊ＝０，１，２，…，ｋ，并且Ｈｎ，ｆｌ在Ｑ（ｂ０（ｌ），…，ｂｋ（ｌ））上不可约（ｌ∈Ｎ）．此外，如果ｘ０是ｆ的周期为ｎ的周期点，则（ｘ－ｘ０）ｘｌ是Ｈｎ，ｆ的一个因子当且仅当（ｘ－ｆ（ｘ０））ｉ是Ｈｎ，ｆ的一个因子．

Let K be a finite extension of the rational number field Q ,and f be a polynomial function with coefficients in K .Suppose that Ω k={α k,f(α k),…,f n-1 (α k)},k=1,2,…,m, are periodic orbits of f of period n ,and denote by H n,f the polynomial whose roots are the periodic points of f of essential period n . Vivaldi and Hatjispyros gave an algorithm to compute the nultipliers of orbits Ω k(k=1,2,…,m ) without computing the orbits themselves under the hypotheses that H n,f is irreducible over K and the multipliers of all orbits Ω k(k=1,2,…,m ) are distinct.We prove in this paper that the hypotheses are unnecessary,that is to say,we will remove the hypotheses and show that the algorthm given by Vivaldi and hatjispyros is still effective.For our purpose we prove that for a given polynomial f=b kx k+…+b 1x+b 0 ,there exists a sequence of polynomials f l=b k(l)x k+…+b 1(l)x+b 0(l)(l∈N) of degree k such that b j(l)-b j(j=0,1,…,k) as l→∞ and H n,f is irreducible over Q(b 0(l),…,b k(l)) for all l∈N . In addition we show that if x 0 is a periodic point of f with essential period n ,then( x-x 0) t is a factor of H n,f only if ( x-f(x 0)) t is a factor of H n,f .