J. Maurice Rojas

A convex geometric approach to counting the roots of a polynomial system

This brief note corrects some errors in the paper quoted in the title, highlights a combinatorial result which may have been overlooked, and points to further improvements in recent literature.

Practical conversion from torsion space to Cartesian space for in silico protein synthesis

Many applications require a method for translating a large list of bond angles and bond lengths to precise atomic Cartesian coordinates. This simple but computationally consuming task occurs ubiquitously in modeling proteins, DNA, and other polymers as well as in many other fields such as robotics. To find an optimal method, algorithms can be compared by a number of operations, speed, intrinsic numerical stability, and parallelization. We discuss five established methods for growing a protein backbone by serial chain extension from bond angles and bond lengths. We introduce the Natural Extension Reference Frame (NeRF) method developed for Rosettas chain extension subroutine, as well as an improved implementation. In comparison to traditional two‐step rotations, vector algebra, or Quaternion product algorithms, the NeRF algorithm is superior for this application: it requires 47% fewer floating point operations, demonstrates the best intrinsic numerical stability, and offers prospects for parallel processor acceleration. The NeRF formalism factors the mathematical operations of chain extension into two independent terms with orthogonal subsets of the dependent variables; the apparent irreducibility of these factors hint that the minimal operation set may have been identified. Benchmarks are made on Intel Pentium and Motorola PowerPC CPUs.

Solving degenerate sparse polynomial systems faster

Consider a system F of n polynomial equations in n unknowns, over an algebraically closed field of arbitrary characteristic. We present a fast method to find a point in every irreducible component of the zero set Z of F. Our techniques allow us to sharpen and lower prior complexity bounds for this problem by fully taking into account the monomial term structure. As a corollary of our development we also obtain new explicit formulae for the exact number of isolated roots of F and the intersection multiplicity of the positive-dimensional part of Z. Finally, we present a combinatorial construction of non-degenerate polynomial systems, with specified monomial term structure and maximally many isolated roots, which may be of independent interest.

Counting Real Connected Components of Trinomial Curve Intersections and m-nomial Hypersurfaces

We prove that any pair of bivariate trinomials has at most five isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees were much larger, e.g., 248832 (for just the non-degenerate roots) via a famous general result of Khovanski. Our bound is sharp, allows real exponents, allows degeneracies, and extends to certain systems of n-variate fewnomials, giving improvements over earlier bounds by a factor exponential in the number of monomials. We also derive analogous sharpened bounds on the number of connected components of the real zero set of a single n-variate m-nomial.

High probability analysis of the condition number of sparse polynomial systems

Let f := (f1 ..... fn) be a random polynomial system with fixed n-tuple of supports. Our main result is an upper bound on the probability that the condition number of f in a region U is larger than 1/e. The bound depends on an integral of a differential form on a toric manifold and admits a simple explicit upper bound when the Newton polytopes (and underlying variances) are all identical.We also consider polynomials with real coefficients and give bounds for the expected number of real roots and (restricted) condition number. Using a Kahler geometric framework throughout, we also express the expected number of roots of f inside a region U as the integral over U of a certain mixed volume form, thus recovering the classical mixed volume when U = (C*)n.

An optimal condition for determining the exact number of roots of a polynomial system

It was shown in ~er75] that the number of roots in (C”) n of a polynomial system depends only on the Newton polytopes of the system, for almost all specializations of the coefficients. This result, henceforth referred to as the BKK bound, gives an upper bound on the number of roots of a polynomial system. The BKK bound is often much better than the Bezout bound for the same system, but the original theorem gives an exact bound only if all the coefficients corresponding to Newton polytope boundaries are generically chosen. In this paper, we show that the BKK bound is exact under much weaker assumptions: only coefficients corresponding to certain vertices of the Newton polytopes need be generic. This result allows application of the BKK bound to many practical problems.

Faster real feasibility via circuit discriminants

We show that detecting real roots for honestly <i>n</i>-variate (<i>n</i>+2)-nomials with integer exponents and coefficients) can be done in time polynomial in the sparse encoding for any fixed n<i>n</i>. The best previous complexity bounds were exponential in the sparse encoding, even for <i>n</i> fixed. We then give a characterization of those functions <i>k</i>(<i>n</i>) such that the complexity of detecting real roots for <i>n</i>-variate(<i>n</i>+<i>k</i>(<i>n</i>))-nomials transitions from <b>P</b> to <b>NP</b>-hardness as <i>n</i> → ∞. Our proofs follow in large part from a new complexity threshold for deciding the vanishing of <i>A</i>-discriminants of <i>n</i>-variate (<i>n</i>+<i>k</i>(<i>n</i>))-nomials. Diophantine approximation, through linear forms in logarithms, also arises as a key tool.

Some Speed-Ups and Speed Limits for Real Algebraic Geometry

We give new positive and negative results, some conditional, on speeding up computational algebraic geometry over the reals: 1.A new and sharper upper bound on the number of connected components of a semi-algebraic set. Our bound is novel in that it is stated in terms of the volumes of certain polytopes and, for a large class of inputs, beats the best previous bounds by a factor exponential in the number of variables. 2.A new algorithm for approximating the real roots of certain sparse polynomial systems. Two features of our algorithm are (a) arithmetic complexity polylogarithmic in the degree of the underlying complex variety (as opposed to the super-linear dependence in earlier algorithms) and (b) a simple and efficient generalization to certain univariate exponential sums. 3.Detecting whether a real algebraic surface (given as the common zero set of some input straight-line programs) is not smooth can be done in polynomial time within the classical Turing model (resp. BSS model over C) only if P=NP (resp. NP?BPP). The last result follows easily from an unpublished observation of S. Smale.

Additive Complexity and Roots of Polynomials over Number Fields and \mathfrak{p} -adic Fields

Consider any nonzero univariate polynomial with rational coefficients, presented as an elementary algebraic expression (using only integer exponents). Letting ?(f) denotes the additive complexity of f, we show that the number of rational roots of f is no more than 15 + ?(f)2(24.01)?(f)?(f)!. This provides a sharper arithmetic analogue of earlier results of Dima Grigoriev and Jean-Jacques Risler, which gave a bound of C?(f)2 for the number of real roots of f, for ?(f) sufficiently large and some constant C with 1<C<32. We extend our new bound to arbitrary finite extensions of the ordinary or p-adic rationals, roots of bounded degree over a number field, and geometrically isolated roots of multivariate polynomial systems. We thus extend earlier bounds of Hendrik W. Lenstra, Jr. and the author to encodings more efficient than monomial expansions. We also mention a connection to complexity theory and note that our bounds hold for a broader class of fields.

Metric Estimates and Membership Complexity for Archimedean Amoebae and Tropical Hypersurfaces

Given any complex Laurent polynomial