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  "body": "\n## Statement\n\nLet $\\mathcal{O}_{n+1} = \\C\\{z_0, \\dots, z_n\\}$ be the [[ring of convergent power series|EVPZ7ZG1HB]].\nFor $f \\in \\mathcal{O}_{n+1}$ with $f(0) = 0$, let the *Tjurina algebra* be\n$$A(f) = \\mathcal{O}_{n+1} \\Big/ \\left( f, \\frac{\\partial f}{\\partial z_0}, \\dots, \\frac{\\partial f}{\\partial z_n} \\right),$$\nthe quotient by the [[ideal|096Z9MP2ME]] generated by $f$ and its partial derivatives.\nLet $f, g \\in \\mathcal{O}_{n+1}$ have [[isolated singularities|VTGYEES3VS]] at $0$.\nThen the following are equivalent:\n\n1. the hypersurface germs $(f^{-1}(0), 0)$ and $(g^{-1}(0), 0)$ are biholomorphic, that is, there are a biholomorphic germ $\\phi$ of $(\\C^{n+1}, 0)$ and a unit $u \\in \\mathcal{O}_{n+1}$ with $g = u \\cdot (f \\circ \\phi)$;\n2. $A(f)$ and $A(g)$ are isomorphic as $\\C$-algebras.\n\n## Remarks\n\nJohn Mather and Stephen Yau proved the theorem in 1982.\nThe direction from 1 to 2 is a direct computation, since the ideal $(f, \\partial f / \\partial z_j)$ is carried to $(g, \\partial g / \\partial z_j)$ by $\\phi$ and $u$.\nThe converse is the hard part.\nIts proof uses the finite determinacy of isolated singularities and a study of the action of the group of coordinate changes on jets.\n\nExample: for $f = x^2 + y^{k+1}$ with $k \\ge 1$, the ideal is $(x, y^k)$, so $A(f) \\cong \\C[y] / (y^k)$.\nThese algebras have different dimensions for different $k$, so the singularities $A_k$ are pairwise not biholomorphic.\n\nThe dimension $\\tau(f) = \\dim_\\C A(f)$ is the *Tjurina number*.\nIt satisfies $\\tau(f) \\le \\mu(f)$, and K. Saito proved that $\\tau(f) = \\mu(f)$ if and only if $f$ is right equivalent to a weighted homogeneous polynomial.\n\n## Explanations\n\n### L1 human\n\nTwo hypersurfaces with an isolated singular point may look different only because of the choice of coordinates.\nThe theorem gives a finite test for whether they are really the same: compute one finite-dimensional algebra for each and compare.\nSo a geometric classification problem becomes a problem about finite-dimensional commutative algebras.\n\n### L2 human\n\nThe Tjurina algebra is also the space of first-order deformations of the hypersurface germ: its dimension $\\tau$ is the number of parameters of the miniversal deformation.\nThe theorem says that this algebra remembers the whole singularity.\nTogether with the [[Milnor number|1ZE51P7GRQ]] and the [[classification of simple singularities|YE5F548WMS]], it is a basic tool of the analytic classification of isolated hypersurface singularities.\n",
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