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Draw the curve y² = x³ and you get something odd: a smooth-looking curve that comes to a sharp point at the origin. Points like that are called singular points, and a large part of algebraic geometry is about understanding and removing them. Shreeram Abhyankar, born in Ujjain, spent much of his life on such questions, and he insisted they could be handled with the kind of algebra a strong school student already knows. This post tells his story and walks through a cusp by hand.
Who was Shreeram Abhyankar?
Abhyankar was born on 22 July 1930 in Ujjain and grew up in Gwalior, where his father taught mathematics, according to the MacTutor biography. He studied at the Royal Institute of Science in Bombay, where he also attended lectures at the Tata Institute of Fundamental Research, and took a BSc in mathematics from Bombay University in 1951.
On the way to Harvard he fell ill on the ship and spent two months recovering in England, so he reached Harvard after the semester had begun. The Purdue mathematics department's tribute describes how he met Oscar Zariski, a leading algebraic geometer, on his first Saturday there. Zariski became his adviser.
What problem did he solve for his PhD?
Zariski mentioned resolution of singularities for surfaces in positive characteristic as an important problem, probably too hard for a PhD. Abhyankar solved it anyway. His PhD came in 1955, and the main result appeared in the Annals of Mathematics in 1956 as "Local Uniformization on Algebraic Surfaces over Ground Fields of Characteristic p ≠ 0".
"Positive characteristic" means arithmetic where adding 1 to itself p times gives 0, as in clock arithmetic modulo a prime p. Many methods that work for ordinary numbers break down there, which is why the problem was hard.
He moved to Purdue in 1963 and was named Marshall Distinguished Professor of Mathematics in 1967. Purdue's tribute notes he also held professorships in computer science and industrial engineering, and wrote over 200 papers. In India he founded the Bhaskaracharya Pratishthan in Pune, according to MacTutor. He died on 2 November 2012 at his desk at home, working on mathematics.
What is a singular point? A worked example
Take the curve y² = x³.
A parametrisation. Put x = t² and y = t³. Then y² = t⁶ and x³ = (t²)³ = t⁶, so every t gives a point on the curve.
Slope by implicit differentiation. Differentiate both sides: 2y (dy/dx) = 3x², so dy/dx = 3x²/(2y).
At the origin. At (0, 0) this becomes 0/0. The formula gives no tangent direction.
Along the parametrisation. dx/dt = 2t and dy/dt = 3t². Both are 0 at t = 0, so the moving point stops dead at the origin before turning back. That is the sharp point, a cusp.
Compare the parabola y = x² with x = t, y = t². Here dx/dt = 1 is never zero, so the curve is smooth everywhere.
What is the Abhyankar-Moh theorem?
In 1975 Abhyankar and T. T. Moh published "Embeddings of the line in the plane" in Crelle's Journal. Roughly, working over the complex numbers, it studies polynomial maps t ↦ (f(t), g(t)) that place a straight line inside the plane with no crossings or sharp points. One consequence: if such a map is a true embedding, then the degree of f divides the degree of g, or the other way round.
Check this against our two curves:
Parabola (t, t²): degrees 1 and 2. 1 divides 2, and indeed the curve is a smooth copy of the line.
Cusp (t², t³): degrees 2 and 3. Neither divides the other, and, as the worked example showed, this curve has a singular point, so it is not an embedding.
The theorem goes far beyond this check, but you can see its flavour with Class 11 tools.
Why did his style stand out?
His colleagues remembered his insistence on concrete working. MacTutor quotes Avinash Sathaye, who wrote that Abhyankar "often rebelled against 'fancy mathematics', demanding that all theorems should have detailed concrete proofs." He also wrote for non-specialists: his article "Historical Ramblings in Algebraic Geometry and Related Algebra" in the American Mathematical Monthly won the Mathematical Association of America's Chauvenet Prize, Purdue's tribute notes.
What can a student learn from Abhyankar?
Work examples by hand. A theorem about curves becomes real once you have parametrised one yourself.
Explain in elementary terms. If you cannot explain a step with simple algebra, you may not understand it yet.
Do not give up on a hard problem because someone says it is too hard.
Read the proofs you use. MacTutor records his rule of not using results whose proofs he had not read.
For another Indian mathematician who worked through examples by the hundred, read our post on Ramanujan's notebooks. If implicit differentiation is still shaky, the Fermi Concept Library covers every concept in mathematics with its own lesson.
Try it: parametrise y² = x²(x + 1) with y = tx, and find the point where the curve crosses itself.
