15 research outputs found
Bounds on the number of real solutions to polynomial equations
We use Gale duality for polynomial complete intersections and adapt the proof
of the fewnomial bound for positive solutions to obtain the bound (e^4+3) 2^(k
choose 2) n^k/4 for the number of non-zero real solutions to a system of n
polynomials in n variables having n+k+1 monomials whose exponent vectors
generate a subgroup of Z^n of odd index. This bound exceeds the bound for
positive solutions only by the constant factor (e^4+3)/(e^2+3) and it is
asymptotically sharp for k fixed and n large.Comment: 5 page
A Special Homotopy Continuation Method For A Class of Polynomial Systems
A special homotopy continuation method, as a combination of the polyhedral
homotopy and the linear product homotopy, is proposed for computing all the
isolated solutions to a special class of polynomial systems. The root number
bound of this method is between the total degree bound and the mixed volume
bound and can be easily computed. The new algorithm has been implemented as a
program called LPH using C++. Our experiments show its efficiency compared to
the polyhedral or other homotopies on such systems. As an application, the
algorithm can be used to find witness points on each connected component of a
real variety
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
Univariate real root isolation over a single logarithmic extension of real algebraic numbers
International audienceWe present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial B â L[x], where L = Q[lg(α)] and α is a positive real algebraic number. The algorithm approximates the coefficients of B up to a sufficient accuracy and then solves the approximate polynomial. For this we derive worst case (aggregate) separation bounds. We also estimate the expected number of real roots when we draw the coefficients from a specific distribution and illustrate our results experimentally. A generalization to bivariate polynomial systems is also presented. We implemented the algorithm in C as part of the core library of mathematica for the case B â Z[lg(q)][x] where q is positive rational number and we demonstrate its efficiency over various data sets