The homotopy type of spaces of real resultants with bounded multiplicity
Let \(\mathbb{K}\) be the field \(\mathbb{R}\) or \(\mathbb{C}\), and let \(d(\mathbb{K})\) be the dimension of \(\mathbb{K}\) as a real vector space. That is \[ d(\mathbb{K})=\begin{cases} 1 &\text{if }\mathbb{K}=\mathbb{R};\\ 2 &\text{if }\mathbb{K}=\mathbb{C}. \end{cases} \] Let \(Q^{d,m}_n(\mathbb{K})\) be the topological space consisting of \(m\)-tuples \((f_1(z),\ldots,f_m(z))\) of degree-\(d\) monic \(\mathbb{K}\)-polynomials such that \(f_1(z),\ldots,f_m(z)\) have no common real root of multiplicity \(\geq n\). In this paper the authors study the homotopy types of the \(Q^{d,m}_n(\mathbb{K})\)'s and identify them as recognizable spaces in low dimensions. To state the main results, let \(\Omega_{[d]_2}\mathbb{RP}^{N}\) be the connected component of \(\Omega\mathbb{RP}^N\) containing null-homotopic maps for \(d\) even, and let it be the other connected component for \(d\) odd; let \(\jmath^{d,m}_{n,\mathbb{K}}\colon Q^{d,m}_n(\mathbb{K})\to\Omega_{[d]_2}\mathbb{RP}^{d(\mathbb{K})mn-1}\) be the jet map given by \[ \jmath^{d,m}_{n,\mathbb{K}}(f_1(z),\ldots,f_m(z))(t)=\begin{cases} [\textbf{f}_1(t):\cdots:\textbf{f}_m(t)] &\text{if }t\in\mathbb{R}\\ [1:\cdots:1] &\text{if }t=\infty, \end{cases} \] where each \(\textbf{f}_i(z)\) is the \(n\)-tuple \[ \textbf{f}_i(z)=(f_1(z),f_1(z)+f^{(1)}_1(z),f_1(z)+f^{(2)}_1(z),\ldots,f_1(z)+f^{(n-1)}_1(z)), \] and let \(D(d;m,n,\mathbb{K})=(d(\mathbb{K})mn-2)\cdot(\lfloor\frac{d}{n}\rfloor+1)-1\) where \(\lfloor x\rfloor\) denotes the integer part of a real number \(x\). Under the assumption \(d(\mathbb{K})mn\geq4\), the authors prove two main theorems (Theorems 1.8 and 1.11): Theorem 1.8 says that \(\jmath^{d,m}_{n,\mathbb{K}}\) is a homotopy equivalence through dimension \(D(d;m,n,\mathbb{K})\), that is the induced group homomorphism \((\jmath^{d,m}_{n,\mathbb{K}})_*\colon\pi_k(Q^{d,m}_n(\mathbb{K}))\to\pi_k(\Omega_{[d]_2}\mathbb{RP}^{d(\mathbb{K})mn-1})\) is an isomorphism for all \(k\leq D(d;m,n,\mathbb{K})\); Theorem 1.11 says that \(Q^{d,m}_n(\mathbb{K})\) is homotopy equivalent to \(J_{\lfloor\frac{d}{n}\rfloor}(\Omega S^{d(\mathbb{K})mn-1})\), the \(\lfloor\frac{d}{n}\rfloor\)-stage James filtration of \(\Omega S^{d(\mathbb{K})mn-1}\). In Section 1 the authors give an introduction and a literature review on this topic, and in Sections 2 and 3 they recall the simplicial resolutions and the Vassiliev spectral sequences that are used to prove the key unstable result (Theorem 4.5) and the stability result (Theorem 5.18). Then they give the proofs of the main theorems, Corollaries 1.9 and 1.10 in Section 6.
scientific article