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  "body": "\n## Summary\n\nDifference algebra studies [[rings|0B1261B5ED]] and [[fields|04C5ZBG1WT]] equipped with a distinguished endomorphism, called difference rings and difference fields.\nIt is the algebraic theory of difference equations, such as $f(x+1) - f(x) = g(x)$, in the same way that differential algebra is the algebraic theory of differential equations.\nThe endomorphism plays the role that the [[derivation|3J89REV9WQ]] plays in a [[differential field|GA5PTQGC6G]].\n\n## Explanations\n\n### L1 human\n\nA difference equation relates the values of an unknown function at shifted points, such as $f(x+1)$ and $f(x)$.\nThe shift $f(x) \\mapsto f(x+1)$ respects sums and products of functions, so it is a ring homomorphism.\nDifference algebra keeps only this structure: a ring together with one such map.\nFor example, on the rational functions $\\C(x)$ the shift sends $x^2$ to $x^2 + 2x + 1$.\nQuestions such as \"can this sum be written in closed form?\" then become questions about difference fields.\n",
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