Complexity of Sparse Polynomial Solving 2: Renormalization
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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, that is on a line integral of the renormalized condition number along all the lifted paths. The theory developed in this paper leads to a continuation algorithm tracking all the solutions between two arbitrary systems of the same structure. The algorithm is randomized, in the sense that it follows a random path between the two systems. It can be modified to succeed with probability one. In order to produce an expected cost bound, several invariants depending solely of the supports of the equations are introduced here. For instance, the mixed surface is a quermassintegral that generalizes surface in the same way that mixed volume generalizes ordinary volume. The face gap measures how close is the set of supporting hyperplanes for a direction in the 0-fan from 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, it is quadratic in the first invariant and linear in the last one.
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