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  "body": "\n## Definition\n\nLet $R$ be a commutative ring, and let $E$ be a [[linear recurrence with constant coefficients|081THG13JZ]] over $R$, of order $d$ with coefficients $c_0, \\dots, c_{d-1}$.\nThe characteristic polynomial of $E$ is the polynomial\n$$\\chi_E(X) = X^d - \\sum_{i=0}^{d-1} c_i X^i \\in R[X].$$\n\n## Remarks\n\nFor the Fibonacci recurrence $u_{n+2} = u_{n+1} + u_n$, the characteristic polynomial is $X^2 - X - 1$. Its roots are the [[golden ratio|0SC8T7548D]] $\\varphi$ and $1 - \\varphi$.\nA geometric sequence $q^n$ is a solution if and only if $q$ is a root of $\\chi_E$. The node [[geometric solutions|096SD4ERQH]] states this.\n\n## Explanations\n\n### L1 human\n\nReplace $u_{n+i}$ by $X^i$ in the recurrence and move everything to one side.\nThis gives the characteristic polynomial, and its roots tell you the geometric sequences that solve the recurrence.\n",
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