Stable multivariate WW-Eulerian polynomials

We prove a multivariate strengthening of Brenti's result that every root of the Eulerian polynomial of type $B$ is real. Our proof combines a refinement of the descent statistic for signed permutations with the notion of real stability-a generalization of real-rootedness to polynomials in multiple variables. The key is that our refined multivariate Eulerian polynomials satisfy a recurrence given by a stability-preserving linear operator. Our results extend naturally to colored permutations, and we also give stable generalizations of recent real-rootedness results due to Dilks, Petersen, and Stembridge on affine Eulerian polynomials of types $A$ and $C$. Finally, although we are not able to settle Brenti's real-rootedness conjecture for Eulerian polynomials of type $D$, nor prove a companion conjecture of Dilks, Petersen, and Stembridge for affine Eulerian polynomials of types $B$ and $D$, we indicate some methods of attack and pose some related open problems.Comment: 17 pages. To appear in J. Combin. Theory Ser.

New Acceleration of Nearly Optimal Univariate Polynomial Root-findERS

Univariate polynomial root-finding has been studied for four millennia and is still the subject of intensive research. Hundreds of efficient algorithms for this task have been proposed. Two of them are nearly optimal. The first one, proposed in 1995, relies on recursive factorization of a polynomial, is quite involved, and has never been implemented. The second one, proposed in 2016, relies on subdivision iterations, was implemented in 2018, and promises to be practically competitive, although user's current choice for univariate polynomial root-finding is the package MPSolve, proposed in 2000, revised in 2014, and based on Ehrlich's functional iterations. By proposing and incorporating some novel techniques we significantly accelerate both subdivision and Ehrlich's iterations. Moreover our acceleration of the known subdivision root-finders is dramatic in the case of sparse input polynomials. Our techniques can be of some independent interest for the design and analysis of polynomial root-finders.Comment: 89 pages, 5 figures, 2 table

Newton's method in practice II: The iterated refinement Newton method and near-optimal complexity for finding all roots of some polynomials of very large degrees

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.Comment: 24 pages, 19 figures. Update in v2 incorporates progress on polynomials of even higher degrees (greater than 1 billion

A Near-Optimal Algorithm for Computing Real Roots of Sparse Polynomials

Let $p\in\mathbb{Z}[x]$ be an arbitrary polynomial of degree $n$ with $k$ non-zero integer coefficients of absolute value less than $2^\tau$. In this paper, we answer the open question whether the real roots of $p$ can be computed with a number of arithmetic operations over the rational numbers that is polynomial in the input size of the sparse representation of $p$. More precisely, we give a deterministic, complete, and certified algorithm that determines isolating intervals for all real roots of $p$ with $O(k^3\cdot\log(n\tau)\cdot \log n)$ many exact arithmetic operations over the rational numbers. When using approximate but certified arithmetic, the bit complexity of our algorithm is bounded by $\tilde{O}(k^4\cdot n\tau)$, where $\tilde{O}(\cdot)$ means that we ignore logarithmic. Hence, for sufficiently sparse polynomials (i.e. $k=O(\log^c (n\tau))$ for a positive constant $c$), the bit complexity is $\tilde{O}(n\tau)$. We also prove that the latter bound is optimal up to logarithmic factors

Computing Real Roots of Real Polynomials

Computing the roots of a univariate polynomial is a fundamental and long-studied problem of computational algebra with applications in mathematics, engineering, computer science, and the natural sciences. For isolating as well as for approximating all complex roots, the best algorithm known is based on an almost optimal method for approximate polynomial factorization, introduced by Pan in 2002. Pan's factorization algorithm goes back to the splitting circle method from Schoenhage in 1982. The main drawbacks of Pan's method are that it is quite involved and that all roots have to be computed at the same time. For the important special case, where only the real roots have to be computed, much simpler methods are used in practice; however, they considerably lag behind Pan's method with respect to complexity. In this paper, we resolve this discrepancy by introducing a hybrid of the Descartes method and Newton iteration, denoted ANEWDSC, which is simpler than Pan's method, but achieves a run-time comparable to it. Our algorithm computes isolating intervals for the real roots of any real square-free polynomial, given by an oracle that provides arbitrary good approximations of the polynomial's coefficients. ANEWDSC can also be used to only isolate the roots in a given interval and to refine the isolating intervals to an arbitrary small size; it achieves near optimal complexity for the latter task.Comment: to appear in the Journal of Symbolic Computatio