Extension in algebraically constructible functions (Q1884150)

Let \(V\subseteq \mathbb R^N\) be a real algebraic set and \(S\subseteq V\) a semialgebraic set. Let \(\varphi\) be a constructible function on \(S\), i.e., an integer-valued function which is constant on each member of some finite semialgebraic partition of \(S\). The author poses the following problem: under which conditions is \(\varphi\) the restriction of an algebraic constructible function on the whole of \(V\)? The ring of algebraic constructible functions on \(V\) has been introduced and studied by McCrory and Parusiński in order to understand the topology of real algebraic sets. An algebraic constructible function is a sum of signs (i.e. \(+1,0,-1\)) of polynomial functions on \(V\). In the paper, the author answers the posed question in the following three cases: a) when \(\varphi\) takes values in \(\{+1,-1\}\), b) when the dimension of \(S\) is less or equal than 2, c) when \(S\) is basic, i.e., \(S\) is described by a conjunction of polynomial inequalities. A first step towards obtaining the desired results is to show that the given problem can be reduced to the generic problem (i.e., to solving the problem up to an algebraic subset of codimension at least one). In particular, it can be assumed that the dimensions of \(V\) and \(S\) are equal. It can also be assumed that \(V\) is irreducible, compact and non-singular, without loss of generality. Then the author gives necessary and sufficient conditions for a constructible function \(\varphi:S \to \mathbb Z\) to be the restriction of an algebraically constructible function on \(V\). These conditions are of the same nature (i.e., that certain constructible functions on certain semialgebraic sets (derived from \(\varphi\) and \(W\)) be the restrictions of some algebraically constructible functions on \(W\)), for each wall \(W\) of \(\varphi\). Since the walls of \(\varphi\) are algebraic subsets of dimension \(\dim V-1\), then the paper also provides an effective method to answer the question for any given such function \(\varphi\). The notions and theorems used in the paper include real spectra, finite fans, quadratic forms and separation of semialgebraic sets.