15 entries matching the current filters. Sorted by recently added.
For n geq 1 fix some sequence of n distinct numbers x 1n , ldots,x nn in [-1,1] . Let epsilon= epsilon(n) to 0 . Does there always exist a continuous function f:[-1,1] to mathbb R such that if p n is a …
Given a 1, ldots,a n in [-1,1] let mathcal L ^nf(x) = sum 1 leq i leq n f(a i) ell i(x) be the unique polynomial of degree n-1 which agrees with f on a i for 1 leq i leq n (that is, the Lagrange inte…
Does there exist a constant c 0 such that, for all large n and all polynomials P of degree n with coefficients pm 1 , max lvert z rvert=1 lvert P(z) rvert (1+c) sqrt n ?
Let C 0 . There exists epsilon 0 such that if n is sufficiently large the following holds. For any x 1, ldots,x n in [-1,1] there exist y 1, ldots,y n in [-1,1] such that, if P is a polynomial of degree $m<(…
For x 1, ldots,x n in [-1,1] let l k(x)= frac prod i neq k (x-x i) prod i neq k (x k-x i) , which are such that l k(x k)=1 and l k(x i)=0 for i neq k . Let x 1,x 2, ldots in [-1,1] be an infinite s…
For x 1, ldots,x n in [-1,1] let l k(x)= frac prod i neq k (x-x i) prod i neq k (x k-x i) , which are such that l k(x k)=1 and l k(x i)=0 for i neq k . What is the minimal value of I(x 1, ldots,x…
Let f(z)= prod i=1 ^n(z-z i) in mathbb C [z] with lvert z i rvert < 1 for all i . Must there always exist a path of length less than 2 in z: lvert f(z) rvert < 1 which connects two of the roots o…
Let f(z)= prod i=1 ^n(z-z i) in mathbb C [z] with lvert z i rvert leq 1 for all i . Let rho(f) be the radius of the largest disc which is contained in z: lvert f(z) rvert< 1 . Determine the behaviour …
Let f in mathbb Z [x] be an irreducible non-constant polynomial such that f(n) geq 1 for all large n in mathbb N . Does there exist a constant c=c(f) 0 such that sum n leq X tau(f(n)) sim cX log X, w…
For any t in (0,1) let t= sum k=1 ^ infty epsilon k(t)2^ -k (where epsilon k(t) in 0,1 ). What is the correct order of magnitude (for almost all t in(0,1) ) for M n(t)= max x in [-1,1] left lvert sum…
Let f(z)= sum 0 leq k leq n epsilon k z^k be a random polynomial, where epsilon k in -1,1 independently uniformly at random for 0 leq k leq n . Is it true that, if R n is the number of roots of f(z) in…
Let ( epsilon k) k geq 0 be independently uniformly chosen at random from -1,1 . If R n counts the number of real roots of f n(z)= sum 0 leq k leq n epsilon k z^k then is it true that, almost surely, …
Let f(z) in mathbb C [z] be a monic non-constant polynomial. Can the set z in mathbb C : lvert f(z) rvert leq 1 be covered by a set of circles the sum of whose radii is leq 2 ?
If p(z) in mathbb C [z] is a monic polynomial of degree n then is the length of the curve z in mathbb C : lvert p(z) rvert=1 maximised when p(z)=z^n-1 ?
Tao proved it for polynomials of sufficiently large degree. The remaining degrees are open.