4 research outputs found
A Polyhedral Homotopy Algorithm For Real Zeros
We design a homotopy continuation algorithm, that is based on numerically
tracking Viro's patchworking method, for finding real zeros of sparse
polynomial systems. The algorithm is targeted for polynomial systems with
coefficients satisfying certain concavity conditions. It operates entirely over
the real numbers and tracks the optimal number of solution paths. In more
technical terms; we design an algorithm that correctly counts and finds the
real zeros of polynomial systems that are located in the unbounded components
of the complement of the underlying A-discriminant amoeba.Comment: some cosmetic changes are done and a couple of typos are fixed to
improve readability, mathematical contents remain unchange
Complexity of Sparse Polynomial Solving 2: Renormalization
Renormalized homotopy continuation on toric varieties is introduced as a tool
for solving sparse systems of polynomial equations, or sparse systems of
exponential sums. The cost of continuation depends on a renormalized condition
length, defined as a line integral of the condition number along all the lifted
renormalized paths.
The theory developed in this paper leads to a continuation algorithm tracking
all the solutions between two generic systems with the same structure. The
algorithm is randomized, in the sense that it follows a random path between the
two systems. The probability of success is one. In order to produce an expected
cost bound, several invariants depending solely of the supports of the
equations are introduced. For instance, the mixed area is a quermassintegral
that generalizes surface area in the same way that mixed volume generalizes
ordinary volume. The facet gap measures for each direction in the 0-fan, how
close is the supporting hyperplane to the nearest vertex. Once the supports are
fixed, the expected cost depends on the input coefficients solely through two
invariants: the renormalized toric condition number and the imbalance of the
absolute values of the coefficients. This leads to a non-uniform complexity
bound for polynomial solving in terms of those two invariants. Up to
logarithms, the expected cost is quadratic in the first invariant and linear in
the last one.Comment: 90 pages. Major revision from the previous versio