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  "body": "\n## Summary\n\nA homogeneous polynomial is a [[polynomial in several variables|07GNX8GCYP]] whose nonzero terms all have the same total degree.\nSuch polynomials are also called forms; the [[quadratic forms|0RBQS1JJCR]] are the homogeneous polynomials of degree $2$.\nA homogeneous polynomial of degree $d$ satisfies $f(\\lambda x) = \\lambda^d f(x)$, so its zero set is a union of lines through the origin.\nThis is why homogeneous polynomials describe the subsets of projective space, and why they appear in the definition of [[quasi-algebraically closed fields|G7RM16N4PF]].\n\n## Explanations\n\n### L1 human\n\nTake a polynomial and add the exponents in each term.\nIf every term gives the same total, the polynomial is homogeneous.\nFor example, $x^3 + 2xy^2 - 5y^3$ is homogeneous of degree $3$, since each term has total degree $3$.\nThe polynomial $x^2 + y$ is not homogeneous, because $x^2$ has degree $2$ and $y$ has degree $1$.\nIf you scale all variables by $2$, a homogeneous polynomial of degree $3$ is scaled by $2^3 = 8$.\n",
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