Dexian’s Post

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We start with #algebraic #geometry today, and I was wondering how best to get cracking, but unwittingly the excellent Elton Papanikolla has provided the perfect material. See below. To solve, let us write x for the height of liquid on the left and y (marked ?) for the height of air on the right. As the bottle is 3/5 full we must have x = (3/2) y. Also comparing total heights we have x+13 = y+17. The resolution of these equations gives us the solution x=12, y=8. You probably did this by algebra, but where has the geometry come in? In this junior school problem we can envisage both equations as straight lines. Generically a pair of lines intersect at a point, giving the answer. But there are some exceptions, and enumerating them all is mildly tedious but here goes: - the equations could be identical (both lines overlap) - they could have no common solution (lines parallel) - one or both equations is a vacuity, 0x + 0y = 0 (not a line but a plane) - one or both equations is an impossibility, 0x + 0y = 1 (empty set) Provided these exceptions don't occur, there is a unique solution and you probably did it by #elimination. Actually, elimination theory is a pre-19th century subject that was overtaken by algebraic geometry in the C20th but has since resurged with the advent of computer algebra. The more abstract algebraic geometry makes a connection between a certain structure known as an #ideal with its solution set or #variety or #manifold. The existence of solutions is then provided by the fantastically-named #Nullstellensatz. This necessitates an introduction to #polynomial #rings, which will be an interesting departure from my usual fare. See you tomorrow!

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