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.