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  "body": "\n## Summary\n\nA Weierstrass polynomial is a monic polynomial in one variable $z_n$ whose other coefficients are holomorphic germs in $z_1, \\dots, z_{n-1}$ vanishing at the origin.\nThe [[Weierstrass preparation theorem|49JSD49P7X]] says that every holomorphic germ that does not vanish identically on the $z_n$-axis is a unit times such a polynomial.\nThis reduces many local questions about holomorphic functions of $n$ variables to questions about polynomials over the ring of germs in $n - 1$ variables.\n\n## Explanations\n\n### L1 human\n\nNear the origin, the zeros of a Weierstrass polynomial of degree $d$ in $z_n$ behave like the $d$ roots of an ordinary polynomial.\nFor each fixed $z' = (z_1, \\dots, z_{n-1})$ near $0$, there are exactly $d$ roots $z_n$, counted with multiplicity, and they all tend to $0$ as $z' \\to 0$.\nFor example, $z_2^2 - z_1$ is a Weierstrass polynomial in $z_2$ of degree $2$, and its roots are the two square roots of $z_1$.\n",
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