PROCDINGS OF TH AMRICAN MATHMATICAL SOCITY Volume 9, Number 8, Pages 83{9 S -9939()588-4 Article electronically published on November 3, AN INQUALITY FOR TH NORM OF A POLYNOMIAL FACTOR IGOR. PRITSKR (Communicated by Albert Baernstein II) Abstract. Let p(z) be a monic polynomial of degree n, with complex coef- cients, and let q(z) be its monic factor. We prove an asymptotically sharp inequality of the form kqk C n kpk,wherekk denotes the sup norm on a compact set in the plane. The best constant C in this inequality is found by potential theoretic methods. We also consider applications of the general result to the cases of a disk and a segment.. Introduction Let p(z) be a monic polynomial of degree n, with complex coecients. Suppose that p(z) has a monic factor q(z), so that p(z) =q(z) r(z) where r(z) is also a monic polynomial. Dene the uniform (sup) norm on a compact set in the complex plane C by (.) kfk = sup jf(z)j z We study the inequalities of the following form (.) kqk C n kpk deg p = n where the main problem is to nd the best (the smallest) constant C,such that (.) is valid for any monic polynomial p(z) and any monic factor q(z). In the case = D, where D = fz jzj < g the inequality (.) was considered in a series of papers by Mignotte [9], Granville [7] and Glesser [6], who obtained anumber of improvements on the upper bound for C D.D.W.Boyd [3] made the nal step here, by proving that (.3) kqk D n kpk D Received by the editors November 8, 999 and, in revised form, November 3, 999. 99 Mathematics Subject Classication. Primary 3C, 3C85 Secondary C8, 3A5. Key words and phrases. Polynomials, uniform norm, logarithmic capacity, equilibrium measure, subharmonic function, Fekete points. Research supported in part by the National Science Foundation grants DMS-99964 and DMS-977359. 83 c American Mathematical Society
84 IGOR. PRITSKR with (.4) = exp =3 log cos t The constant = C D is asymptotically sharp, as n and it can also be expressed in a dierent way, using Mahler's measure. This problem is of importance in designing algorithms for factoring polynomials with integer coecients over integers. We refer to [5] and [8] for more information on the connection with symbolic computations. A further development related to (.) for =[;a a] a > was suggested by P. B. Borwein in [] (see Theorems and 5 there or see Section 5.3 in []). In particular, Borwein proved that if deg q = m, then Y m (.5) jq(;a)j kpk [;a a] a m;n n; +cos k ; n k= where the bound is attained for a monic Chebyshev polynomial of degree n on [;a a] and a factor q. He also showed that, for =[; ], the constant inthe above inequality satises which hints that (.6) sup n m; @ [n=3]; n =3 = exp C [; ] =exp =3 my k= [n=3] + cos k ; n =n Y k= + cos k ; n log ( + cos x) dx log ( + cos x) dx =n A =98 =98 We nd the asymptotically best constant C in (.) for a rather arbitrary compact set. The general result is then applied to the cases of a disk and a line segment, so that we recover (.3){(.4) and conrm (.6).. Results Our solution of the above problem is based on certain ideas from the logarithmic potential theory (cf. [] or [3]). Let cap() bethe logarithmic capacity of a compact set C. For with cap() >, denote the equilibrium measure of (in the sense of the logarithmic potential theory) by.weremark that is a positive unit Borel measure supported on, supp (see [3, p.55]). Theorem.. Let C be acompact set, cap() >. Then the best constant C in (.) is given by (.) C = max exp log jz ; ujd (z) u@ jz;uj cap()
AN INQUALITY FOR TH NORM OF A POLYNOMIAL FACTOR 85 Furthermore, if is regular, then (.) C =max exp ; log jz ; ujd (z) u@ jz;uj The above notion of regularity is to be understood in the sense of the exterior Dirichlet problem (cf. [3, p. 7]). Note that the condition cap() > is usually satised for all applications, as it only fails for very thin sets (see [3, pp. 63{66]), e.g., nite sets in the plane. But if consists of nitely many points, then the inequality (.) cannot be true for a polynomial p(z) with zeros at every point of and for its linear factors q(z). On the other hand, Theorem. is applicable to any compact set with a connected component consisting of more than one point (cf. [3, p. 56]). One can readily see from (.) or (.) that the best constant C is invariant under the rigid motions of the set in the plane. Therefore we consider applications of Theorem. to the family of disks D r = fz jzj <rg, which are centered at the origin, and to the family of segments [;a a] a > Corollary.. Let D r be a disk of radius r. Then the best constant C Dr, for = D r, is given by (.3) C Dr = 8 >< > <r = r r exp ; arcsin r log r cos x dx r>= Note that (.3){(.4) immediately follow from (.3) for r = The graph of C Dr, as a function of r, is in Figure..8.6.4. 4 6 8 Figure. C Dr as a function of r. Corollary.3. If =[;a a] a >, then (.4) C [;a a] = 8 >< > <a= a a a exp ;a log(t p + a) a ; t a>=
86 IGOR. PRITSKR Observe that (.4), with a =, implies (.6) by the change of variable t = cos x We include the graph of C [;a a], as a function of a, in Figure. 4 3.5 3.5.5 4 6 8 Figure. C [;a a] as a function of a. We now state two general consequences of Theorem.. They explain some interesting features of C,which the reader may have noticed in Corollaries. and.3. Let be the uclidean diameter of. diam() = max jz ; j z Corollary.4. Suppose that cap() > If diam(), then (.5) C = cap() It is well known that cap(d r )=r and cap([;a a]) = a= (see [, p. 35]), which claries the rst lines of (.3) and (.4) by (.5). The next corollary shows how the constant C behaves under dilations of the set. Let be the dilation of with a factor > Corollary.5. If is regular, then (.6) + C = Thus Figures and clearly illustrate (.6). We remark that one can deduce inequalities of the type (.), for various L p norms, from Theorem., by using relations between L p and L norms of polynomials on (see, e.g., []). 3. Proofs Proof of Theorem.. The proof of this result is based on the ideas of [3] and []. For u C, consider a function u (z) = max(jz ; uj ) z C One can immediately see that log u (z) is a subharmonic function in z C, which has the following integral representation (see [, p. 9]) (3.) log u (z) = log jz ; tj d u (t) z C
AN INQUALITY FOR TH NORM OF A POLYNOMIAL FACTOR 87 where d u (u + e i )=d=() is the normalized angular measure on jt ; uj =. Let u @ be such that kqk = jq(u)j If z k k= n are the zeros of p(z), arranged so that the rst m zeros belong to q(z), then (3.) log kqk = = mx nx k= k= log ju ; z k j mx k= log jz k ; tj d u (t) = log u (z k ) nx k= log u (z k ) log jp(t)j d u (t) by (3.). We use the well known Bernstein-Walsh lemma about the growth of a polynomial outside of the set (see [, p. 56], for example) Let C be a compact set, cap() >, with the unbounded component ofc n denoted by. Then, for any polynomial p(z) of degree n, we have (3.3) jp(z)j kpk e ng(z ) z C where g (z ) is the Green function of, with pole at. The following representation for g (z ) is found in Theorem III.37 of [3, p. 8]). (3.4) g (z ) =log cap() + log jz ; tj d (t) z C It follows from (3.){(3.4) and Fubini's theorem that n log kqk kpk log jp(t)j=n kpk =n d u (t) g (t ) d u (t) = log cap() + log jz ; tj d u (t)d (z) = log cap() + log u (z) d (z) Using the denition of u (z), we obtain from the above estimate that n max u@ kqk B exp log u (z) d (z) @ C cap() A kpk = B @ max u@ exp jz;uj cap() log jz ; uj d (z) C A n kpk Hence (3.5) C max exp log jz ; uj d (z) u@ jz;uj cap()
88 IGOR. PRITSKR In order to prove the inequality opposite to (3.5), we consider the n-th Fekete points fa k n g n k= for the set (cf. [, p. 5]). Let p n (z) = ny k= (z ; a k n ) be the Fekete polynomial of degree n. Dene the normalized counting measures on the Fekete points by n = n nx k= ak n n N It is known that (see Theorems 5.5.4 and 5.5. in [, pp. 53{55]) (3.6) n kp nk =n =cap() Furthermore, we have the following weak* convergence of counting measures (cf. [, p. 59]) (3.7) n as n Let u @ be a point, where the maximum on the right-hand side of (3.5) is attained. Dene the factor q n (z) forp n (z), with zeros being the n-th Fekete points satisfying ja k n ; uj. Then we have by (3.7) that n kq nk =n n jq n(u)j =n = exp @ n n = exp n = exp jz;uj jz;uj X ja k n;uj log ju ; zj d n (z) log ju ; zjd (z) log ju ; a k n j Combining the above inequality with (3.6) and the denition of C,we obtain that exp kq n k =n C n kp n k =n jz;uj log jz ; uj d (z) cap() This shows that (.) holds true. Moreover, if u @ is a regular point for, then we obtain by Theorem III.36 of [3, p. 8]) and (3.4) that log cap() + log ju ; tj d (t) =g (u ) = Hence log cap() jz;uj + log ju ; tj d (t) =; log ju ; tj d (t) jz;uj which implies (.) by (.). A
AN INQUALITY FOR TH NORM OF A POLYNOMIAL FACTOR 89 Proof of Corollary.. It is well known [3, p. 84] that cap(d r )=r and d Dr (re i ) = d=(), where d is the angular measure on @D r. If r ( =], then the numerator of (.) is equal to, so that C Dr = <r = r Assume that r>= We setz = re i and let u = re i be a point where the maximum in (.) is attained. On writing we obtain that C Dr = r exp = r exp = r exp jz ; u j =r sin ; +; arcsin r + arcsin r ; arcsin r arcsin r ; log ; arcsin r by the change of variable ; = ; x log r sin ; r cos x dx log r cos x dx d Proof of Corollary.3. Recall that cap([;a a]) = a= (see [3, p.84])and It follows from (.) that (3.8) C [;a a] = a exp d [;a a] (t) = p a ; t t [;a a] max log p jt ; uj u[;a a] [;a a]n(u; u+) a ; t If a ( =], then the integral in (3.8) obviously vanishes, so that C [;a a] ==a. For a>=, let (3.9) f(u) = One can easily see from (3.9) that and f (u) = f (u) = a u+ u; ;a However, if u ( ; a a ; ), then f (u) = a u+ [;a a]n(u; u+) (u ; t) p a ; t < (u ; t) p a ; t > log jt ; uj p a ; t u [;a ; a] u [a ; a] u; (u ; t) p a ; + t (u ; t) p a ; t ;a
9 IGOR. PRITSKR It is not dicult to verify directly that (u ; t) p a ; t = p a ; u log a which implies that f (u) = p a ; u log for u ( ; a a ; ) Hence ; ut + p a ; p t a ; u t ; u + C p a ; u + u + a ; (u ; ) p p a ; u a ; u ; u + a ; (u +) p a ; u f (u) < u ( ; a ) and f (u) > u ( a; ) Collecting all facts, we obtain that the maximum for f(u) on[;a a] is attained at the endpoints u = a and u = ;a, and it is equal to a max f(u) = log(t p + a) u[;a a] ;a a ; t Thus (.3) follows from (3.8) and the above equation. Proof of Corollary.4. Note that the numerator of (.) is equal to, because jz ; uj 8 z 8 u @. Thus (.5) follows immediately. Proof of Corollary.5. Observe that C for any C, so that C. Since is regular, we use the representation for C in (.). Let T be the dilation mapping. Then jtz;tuj = jz;uj z u and d (Tz)=d (z). This gives that C = max exp ; log jtz; Tujd (Tz) Tu@() jtz;tuj = max exp ; log(jz ; uj) d (z) u@ jz;uj= = max u@ exp ; (D = (u)) log ; jz;uj= < max exp ; log jz ; uj d (z) u@ jz;uj= log jz ; uj d (z) where. Using the absolute continuity of the integral, we have that which implies (.6). + jz;uj= log jz ; uj d (z) = References [] P. B. Borwein, xact inequalities for the norms of factors of polynomials, Can. J. Math. 46 (994), 687-698. MR 95k65 [] P. Borwein and T. rdelyi, Polynomials and Polynomial Inequalities, Springer-Verlag, New York, 995. MR 97e4 [3] D. W. Boyd, Two sharp inequalities for the norm of a factor of a polynomial, Mathematika 39 (99), 34-349. MR 94a6 [4] D. W. Boyd, Sharp inequalities for the product of polynomials, Bull. London Math. Soc. 6 (994), 449-454. MR 95m38
AN INQUALITY FOR TH NORM OF A POLYNOMIAL FACTOR 9 [5] D. W. Boyd, Large factors of small polynomials, Contemp. Math. 66 (994), 3-38. MR 95e34 [6] P. Glesser, Nouvelle majoration de la norme des facteurs d'un polyn^ome, C. R. Math. Rep. Acad. Sci. Canada (99), 4-8. MR 9d [7] A. Granville, Bounding the coecients of a divisor of a given polynomial, Monatsh. Math. 9 (99), 7-77. MR 9g [8] S. Landau, Factoring polynomials quickly, Notices Amer. Math. Soc. 34 (987), 3-8. MR 87k [9] M. Mignotte, Some useful bounds, In \Computer Algebra, Symbolic and Algebraic Computation" (B. Buchberger et al., eds.), pp. 59-63, Springer-Verlag, New York, 98. CMP 66 [] I.. Pritsker, Products of polynomials in uniform norms, to appear in Trans. Amer. Math. Soc. [] I.. Pritsker, Comparing norms of polynomials in one and several variables, J. Math. Anal. Appl. 6 (997), 685-695. MR 93j3 [] T. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, Cambridge, 995. MR 96e3 [3] M. Tsuji, Potential Theory in Modern Function Theory, Chelsea Publ. Co., New York, 975. MR 5499 Department of Mathematics, 4 Mathematical Sciences, Oklahoma State University, Stillwater, Oklahoma 7478-58 -mail address igor@math.okstate.edu