Let Qn(x) = ∑n i=0 Aix i be a random algebraic polynomial where the coefficients A0, A1, · · · form a sequence of centered Gaussian random variables. Moreover, assume that the increments ∆j = Aj−Aj−1, j = 0, 1, 2, · · · are independent, assuming A−1 = 0. The coefficients can be considered as n consecutive observations of a Brownian motion. We obtain the asymptotic behaviour of the expected numb...

Numerous problems in numerical analysis, including matrix inversion, eigenvalue calculations and polynomial zerofinding, share the following property: The difficulty of solving a given problem is large when the distance from that problem to the nearest "ill-posed" one is small. For example, the closer a matrix is to the set of noninvertible matrices, the larger its condition number with respect...

with derivative g b(w) = w n−1 , and use the conjugacy invariant b̂ = b . These normal forms (0.1) and (0.2) are related by the change of variable formula w = nz with b̂ = nĉ , and hence b = nc . (In particular, in the degree two case, b = b̂ is equal to 4c = 4ĉ .) If A is any ring contained in the complex numbers C , it will be convenient to use the non-standard notation A for the integral closur...

This paper provides an asymptotic estimate for the expected number of real zeros of a random algebraic polynomial a0 + a1x + a2x + ···+ an−1xn−1. The coefficients aj ( j = 0,1,2, . . . ,n− 1) are assumed to be independent normal random variables withmean zero. For integers m and k = O(logn)2 the variances of the coefficients are assumed to have nonidentical value var(aj) = ( k−1 j−ik ) , where ...

Numerically Solving Polynomial Systems with Bertini • approaches numerical algebraic geometry from a user's point of view with many worked examples, • teaches how to use Bertini and includes a complete reference guide, • treats the fundamental task of solving a given polynomial system and describes the latest advances in the field, including algorithms for intersecting and projecting algebraic ...

The first essential ingredient to build up Stein’s method for a continuous target distribution is identify so-called Stein operator, namely linear differential operator with polynomial coefficients. In this paper, we introduce the notion of algebraic operators (see Definition 3.4), and provide novel find all given order degree random variable form Y=h(X), where X=(X1,…,Xd) has i.i.d. standard G...

In this paper, I will prove that assuming Schanuel’s conjecture, an exponential polynomial with algebraic coefficients can have only finitely many algebraic roots. Furthermore, this proof demonstrates that there are no unexpected algebraic roots of any exponential polynomial. This implies a special case of Shapiro’s conjecture: if p(x) and q(x) are two exponential polynomials over the algebraic...