Interpolation and the Chinese Remainder Theorem

We have readers of all backgrounds here at Ars Math, so I thought I would experiment with a more expository post than usual. Commenter Wendell Dryden is teaching himself the Chinese remainder theorem.

The result has been considerably generalized (as the Wikipedia entry makes clear), and one variant is easier to understand (I think): polynomial interpolation. Given n numbers xi on the x-axis, and n numbers, yi, on the y-axis, you can always find a polynomial p such that p(xi) = yi. The steps in solving this problem and the integer congruence problem are similar, and both problems can be solved by using the extended Euclidean algorithm.

This analogy has been taken much further in algebraic geometry. From that point of view, an integer is no longer just a number, but actually (like a polynomial) a function. The integer, in its function guise, sends prime numbers to that integer modulo that prime. So the function 67 sends 2 to 1, 3 to 1, 5 to 2, 7 to 4, 11 to 1, etcetera. (The value eventually stabilizes, in this instance at 67, which always seemed to me must be a fact of some significance, but I’ve never seen it used for anything.)

So now you know: to an algebraic geometer, integers are functions. Mathematics is like drugs, but cheaper.

6 thoughts on “Interpolation and the Chinese Remainder Theorem

  1. Isn’t the fact that the value eventually stabilizes what makes the integer an integer instead of some funny hypernatural, or p-adic thing? And of course, there must be some funny way you can glue these functions together to get a scheme that’s locally a lot like Spec(Z) but got some funny global behavior. At least, that’s what I remember from my first year algebraic geometry class.

  2. Hmmm… Frankly, I don’t think Wendell’s teaching himself very much. But the X-Y reference… that’s, helpful. I can see how that fits. It would also be helpful if Wendell learned some math that deals with sets. (This is a guy who still finds division exciting and mysterious.)

    :)

  3. The p-adics can only be reduced mod p^n, not by any other prime. The hypernaturals are close to what I had in mind, but they of course stabilize eventually, just after an infinite number of steps. :-)

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