What is an algebraic variety?

It’s the answers to an equation.

Why might you care? See the sidebar. →


A 1-dimensional variety.

Actually, it can be either one polynomial equation or a simultaneous system of polynomial equations.

Each polynomial can have more than one letter so you don’t just have options like 3xxx − 4xx + 5x = 0 but also like abbc + 2aaa − bbbcccc = 0. But each equation should have finitely many terms, so you can’t for example sneak sinewaves in by infinite approximation.

(x²−y²−1) • (x²−z²−1) • (y²−z²−1) = 0


x²⁶•y + x•z⁶ + y¹³•z + x⁹•y¹³ + z²⁶ = 0


(all pictures, 1-letter and 3-letter, from isomorphismes)





Changing the =0 on the RHS to =5 or some other number would shift the horizontal bar up and down, “selecting” a different variety from the same curve.

somewhere, someone is doing the limbo
a little lower
a little bit higher now


It’s conventional to move everything over to one side so x³ − x² + x − 5 would be the name of the variety, instead of { ∀ x | x³ − x² + x = 5 }.

Originally published on isomorphism.es.