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  "body": "\n## Statement\n\nLet $k$ be a field and let $x = (x_1, \\ldots, x_n)$ be variables.\nLet $k[\\![x]\\!]$ be the [[ring of formal power series|3E8ZG3NJYZ]] in $x$.\nLet $k\\langle x \\rangle \\subseteq k[\\![x]\\!]$ be the ring of algebraic power series: those that are algebraic over the [[polynomial ring|0SDC934R52]] $k[x]$.\nLet $y = (y_1, \\ldots, y_N)$ be further variables and let $f_1, \\ldots, f_m \\in k[x, y]$ be polynomials.\nAssume that $\\hat{y} = (\\hat{y}_1, \\ldots, \\hat{y}_N) \\in k[\\![x]\\!]^N$ satisfies $f_j(x, \\hat{y}) = 0$ for $j = 1, \\ldots, m$.\nThen for every integer $c \\ge 1$ there is $\\bar{y} \\in k\\langle x \\rangle^N$ such that $f_j(x, \\bar{y}) = 0$ for $j = 1, \\ldots, m$ and $\\bar{y}_i \\equiv \\hat{y}_i \\pmod{(x_1, \\ldots, x_n)^c}$ for $i = 1, \\ldots, N$.\n\n## Remarks\n\nThe ring $k\\langle x \\rangle$ is the [[Henselization|917PGHVZPM]] of the local ring of $k[x]$ at the ideal $(x_1, \\ldots, x_n)$.\nThe theorem says that formal solutions of polynomial equations can be approximated, in the [[$(x)$-adic topology|2VF2TSJ3PY]], by algebraic solutions.\nM. Artin proved it in 1969.\nPopescu's theorem extends the result to every excellent Henselian local ring in place of $k\\langle x \\rangle$.\n\nExample: the equation $y_1 - y_2 = 0$ has the formal solution $\\hat{y}_1 = \\hat{y}_2 = \\sum_{i \\ge 0} x^i / i!$ over $\\Q$, which is not algebraic.\nThe truncations $\\bar{y}_1 = \\bar{y}_2 = \\sum_{i \u003c c} x^i / i!$ are polynomials, hence algebraic, and agree with $\\hat{y}$ modulo $(x)^c$.\nThe theorem does not say that the formal solution itself is algebraic; it only gives algebraic solutions that agree with it to any given order.\n\n## Explanations\n\n### L1 human\n\nSuppose a system of polynomial equations has a solution made of formal power series.\nSuch a solution may be wild: the series may not come from any algebraic equation.\nThe theorem says that there are tame solutions nearby, made of algebraic power series, which agree with the formal solution up to any order you choose.\nSo questions about formal solutions can often be moved to the much smaller world of algebraic functions.\n\n### L2 human\n\nIn algebraic geometry, formal power series describe infinitesimal neighbourhoods, while algebraic power series describe étale neighbourhoods.\nThe theorem lets one pass from a formal statement about the completion of a local ring to a statement in the [[Henselization|917PGHVZPM]], and so to an étale neighbourhood.\nThis is used, for example, to compare the formal and the étale local structure of singularities.\nIt is also a key input in Artin's work on algebraic spaces and moduli problems.\n",
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