Abstract. We relate the monodromy group of a branched cover to the monodromy groups of its restrictions. We show that the presence of a transposition in a restricted monodromy group significantly constrains the monodromy group of the original branched cover. Using this result, we complete the sparse trace test algorithm, which numerically verifies whether a set of solutions to a sparse polynomial system is complete. Along the way, we develop a generalization of the notion of a trace test in the setting of branched covers.

Abstract. We prove there are 1442 four-bar coupler curves through nine generic points in the plane, and thus resolve Alt’s problem. We obtain this proof in three steps. First, we identify the space of coupler curves with a Zariski open subset of Gr(3, 6). Next, we formulate the polynomial system representing the nine-point path synthesis problem in these coordinates and modify it to obtain the mixed volume 5538. Finally, we prove that 4096 of the branches of the generic sparse polynomial system with that support escape the torus in the sparse limit. Thus, we obtain an upper bound of 5538 − 4096 = 1442 for the generic solution count. A lower bound of 1442 is achieved via numerical certification on one instance

Abstract. We give an algorithm, based on second-order necessary conditions, to test whether a point is isolated on an algebraic set. Using this subroutine, we develop an algorithm that bounds the local dimension of an algebraic set at a point. The principal computation in our isolation test is governed by the nullity of the Jacobian, avoiding the combinatorial growth of classical methods. For exact input, the bound returned by the algorithm is certified. We compare our approach with classical methods. As an application, we show that the realization space of the 24-cell has the expected local dimension at its symmetric regular realization, completing the local dimension program for regular 4-polytopes initiated by Rastanawi, Sinn, and Ziegler. 

Abstract. We exhibit three conics with 144 tritangent circles, disproving the conjectured maximum of 136.