polynomial.funext

Stream: maths

[Johan Commelin]

Under what conditions on a ring R can I conclude that if a (multivariate) polynomial over R evaluates to zero on all x \in R (or tuples (x_0, .., x_n, ..) in the mv case) that it is the zero polynomial.

• For finite R it's of course false: frobenius
• For infinite integral domains I can prove it.
• After that, you can take products of such rings.
• But I'm not so sure about what happens if R has a nilradical. Can anyone help me out?

[Bryan Gin-ge Chen]

This MO question seems related: https://mathoverflow.net/questions/160986/rings-for-which-no-polynomial-induces-the-zero-function

[Johan Commelin]

@Bryan Gin-ge Chen thanks! I'll just stick to infinite integral domains then. Because those answers aren't very clean (-;

[Kevin Buzzard]

Oh we ran into exactly this question when formalising the basics of algebraic varieties. Our source said k an algebraically closed field, which worked because such things are infinite integral domains :-)

[Kenny Lau]

Frobenius stands as a counter-example to Fp, but for general finite R = {r1, ..., rn} one can consider (X - r1) ... (X - rn) as a counter-example

[Johan Commelin]

Sure

[Johan Commelin]

#4196