Abstract. We use Newton's method to find all roots of several polynomials in one complex variable of degree up to and exceeding one million and show that the method, applied to appropriately chosen starting points, can be turned into an algorithm that can be applied routinely to find all roots without deflation and with the inherent numerical stability of Newton's method.We specify an algorithm that provably terminates and finds all roots of any polynomial of arbitrary degree, provided all roots are distinct and exact computation is available. It is known that Newton's method is inherently stable, so computing errors do not accumulate; we provide an exact bound on how much numerical precision is sufficient.

We present a practical implementation based on Newton's method to find all roots of several families of complex polynomials of degrees exceeding one billion (10 9 ) so that the observed complexity to find all roots is between O(d ln d) and O(d ln 3 d) (measuring complexity in terms of number of Newton iterations or computing time). All computations were performed successfully on standard desktop computers built between 2007 and 2012.

We show that modular operads are equivalent to modules over a certain simple properad which we call the Brauer properad. Furthermore we show that, in this setting, the Feynman transform corresponds to the cobar construction for modules of this kind. To make this precise, we extend the machinery of the bar and cobar constructions relative to a twisting morphism to modules over a general properad. This generalizes the classical case of algebras over an operad and might be of independent interest. As an application, we sketch a Koszul duality theory for modular operads.

In this paper we study versions of Goodwillie's calculus of functors that use diagrams other than cubes. We first prove that the universal excisive approximation exists for a larger class of diagrams, which we call shapes. We then study the extension of the Taylor tower that contains all of these approximations. We prove that its limit agrees with the limit of the tower using certain criteria for the existence of maps between two excisive approximations. Lastly we investigate the question when a shape yields a notion of excision equivalent to one of the classical ones.

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