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  "body": "\n## Statement\n\nLet $f(x_0, \\dots, x_n)$ and $g(y_0, \\dots, y_m)$ be holomorphic functions near the origin, each with an [[isolated singularity|VTGYEES3VS]] at $0$.\nLet $f \\oplus g$ be the function $(x, y) \\mapsto f(x) + g(y)$ near $0$ in $\\C^{n+1} \\times \\C^{m+1}$; it also has an isolated singularity at $0$.\nLet $F_f$, $F_g$ and $F_{f \\oplus g}$ be the Milnor fibres of the [[Milnor fibrations|KTFG1S2SMG]] at $0$, with [[monodromies|C8HD62AB6Q]] $T_f$, $T_g$ and $T_{f \\oplus g}$ on [[reduced homology|NJQMWVK3QK]].\nThen there is an isomorphism\n$$\\tilde H_{n+m+1}(F_{f \\oplus g}; \\Z) \\cong \\tilde H_n(F_f; \\Z) \\otimes \\tilde H_m(F_g; \\Z)$$\nunder which $T_{f \\oplus g} = T_f \\otimes T_g$.\n\n## Remarks\n\nSebastiani and Thom proved the theorem in 1971.\nThe idea is that $F_{f \\oplus g}$ has the homotopy type of the join $F_f * F_g$.\nThe reduced homology of a join is the tensor product of the reduced homologies, shifted by one degree.\nThe monodromy of $f \\oplus g$ is the join of the two monodromies.\n\nTaking ranks gives $\\mu(f \\oplus g) = \\mu(f) \\, \\mu(g)$ for the [[Milnor numbers|1ZE51P7GRQ]].\nThis also follows from algebra, since the local algebra of $f \\oplus g$ is the tensor product of the local algebras of $f$ and $g$.\n\nExample: $f(x) = x^p$ and $g(y) = y^q$.\nThe Milnor fibre of $x^p$ is $p$ points, cyclically permuted by the monodromy, whose eigenvalues on reduced $H_0$ are the $p$-th roots of unity other than $1$.\nSo the monodromy of $x^p + y^q$ on $H_1$ has as eigenvalues the products $\\zeta \\eta$ with $\\zeta^p = 1 \\ne \\zeta$ and $\\eta^q = 1 \\ne \\eta$, and $\\mu(x^p + y^q) = (p-1)(q-1)$.\n\n## Explanations\n\n### L1 human\n\nIf a function is a sum of two functions in separate sets of variables, its singularity is built from the two singularities.\nThe theorem says how: the topology of the sum is a product, in a precise sense, of the topologies of the summands.\nFor instance, the Milnor number of the sum is the product of the two Milnor numbers.\n\n### L2 human\n\nThe theorem reduces many computations to one-variable functions.\nIterating it, for a Brieskorn–Pham polynomial $z_0^{a_0} + \\cdots + z_n^{a_n}$, the monodromy is diagonalizable with eigenvalues $\\zeta_0 \\cdots \\zeta_n$, where $\\zeta_j^{a_j} = 1 \\ne \\zeta_j$, and $\\mu = \\prod_j (a_j - 1)$.\nThis is the computation behind the topology of [[Brieskorn manifolds|5E9E5R9RYZ]].\nAdding a square $z^2$ in a new variable does not change $\\mu$, and it changes the monodromy only by a sign.\nThis explains the stable behaviour of singularities under adding squares.\n",
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