On hardness of computing analytic Brouwer degree

We prove that counting the analytic Brouwer degree of rational coefficient polynomial maps in $\operatorname{Map}(\mathbb C^d, \mathbb C^d)$ -- presented in degree-coefficient form -- is hard for the complexity class $\operatorname{\sharp P}$, in the following sense: if there is a randomized polynomial time algorithm that counts the Brouwer degree correctly for a good fraction of all input instances (with coefficients of bounded height where the bound is an input to the algorithm), then $\operatorname{P}^{\operatorname{\sharp P}} =\operatorname{BPP}$

On Computability and Triviality of Well Groups

The concept of well group in a special but important case captures homological properties of the zero set of a continuous map $f:K\to R^n$ on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within $L_\infty$ distance r from f for a given r>0. The main drawback of the approach is that the computability of well groups was shown only when dim K=n or n=1. Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of R^n by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and dim K<2n-2, our approximation of the (dim K-n)th well group is exact. For the second part, we find examples of maps $f,f': K\to R^n$ with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status.Comment: 20 pages main paper including bibliography, followed by 22 pages of Appendi

LIPIcs

The concept of well group in a special but important case captures homological properties of the zero set of a continuous map f from K to R^n on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within L_infty distance r from f for a given r &gt; 0. The main drawback of the approach is that the computability of well groups was shown only when dim K = n or n = 1. Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of R^n by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and dim K &lt; 2n-2, our approximation of the (dim K-n)th well group is exact. For the second part, we find examples of maps f, f' from K to R^n with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status

Persistence of Zero Sets

We study robust properties of zero sets of continuous maps $f:X\to\mathbb{R}^n$. Formally, we analyze the family $Z_r(f)=\{g^{-1}(0):\,\,\|g-f\|<r\}$ of all zero sets of all continuous maps $g$ closer to $f$ than $r$ in the max-norm. The fundamental geometric property of $Z_r(f)$ is that all its zero sets lie outside of $A:=\{x:\,|f(x)|\ge r\}$. We claim that once the space $A$ is fixed, $Z_r(f)$ is \emph{fully} determined by an element of a so-called cohomotopy group which---by a recent result---is computable whenever the dimension of $X$ is at most $2n-3$. More explicitly, the element is a homotopy class of a map from $A$ or $X/A$ into a sphere. By considering all $r>0$ simultaneously, the pointed cohomotopy groups form a persistence module---a structure leading to the persistence diagrams as in the case of \emph{persistent homology} or \emph{well groups}. Eventually, we get a descriptor of persistent robust properties of zero sets that has better descriptive power (Theorem A) and better computability status (Theorem B) than the established well diagrams. Moreover, if we endow every point of each zero set with gradients of the perturbation, the robust description of the zero sets by elements of cohomotopy groups is in some sense the best possible (Theorem C)

On Computability and Triviality of Well Groups

The concept of well group in a special but important case captures homological properties of the zero set of a continuous map f from K to R^n on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within L_infty distance r from f for a given r > 0. The main drawback of the approach is that the computability of well groups was shown only when dim K = n or n = 1. Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of R^n by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and dim K < 2n-2, our approximation of the (dim K-n)th well group is exact. For the second part, we find examples of maps f, f\u27 from K to R^n with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status