Birch's theorem

Birch's theorem gives conditions under which systems of homogeneous forms have an asymptotic formula for the number of integer solutions of bounded size.

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Birch rank: The codimension of the locus where the Jacobian matrix of a system of forms has less than full rank. Birch's theorem uses this rank measure to set a threshold for its asymptotic conclusion.
Homogeneous polynomial: A polynomial whose monomials all have the same total degree. Birch's theorem applies to systems of forms, meaning homogeneous polynomials.
Bryan Birch: A British mathematician known for work in number theory, including the theorem on integer solutions of systems of forms. His name is attached to the theorem and its foundational proof.
Hasse principle: A local-global principle asserting that rational solvability can be tested over the real and p-adic fields. Birch's local factors reflect these local tests, though its asymptotic conclusion needs stronger hypotheses.
Davenport's inequality: A bound relating the number of integer solutions of polynomial equations to the size of their coefficients and variables. It gives upper bounds in settings where Birch's theorem can provide a full asymptotic.
Jacobian matrix: A matrix of first partial derivatives that describes the linear approximation of a differentiable map. Its rank detects singular points of the system of forms.
Diophantine equation: An equation whose solutions are sought among integers or other specified number systems. The theorem counts integer solutions to a family of Diophantine equations.
Hardy–Littlewood circle method: An analytic method developed by G. H. Hardy and J. E. Littlewood for estimating integer solutions of equations. Birch extended its use from individual equations to systems of forms.
Hardy–Littlewood singular series: An Euler product of local factors that adjusts a circle-method main term for arithmetic obstructions. It is a component of the leading constant in Birch-type asymptotic formulas.
Schmidt's subspace theorem: A theorem in Diophantine approximation restricting integer points that satisfy unusually strong linear-form inequalities. It controls exceptional solutions by approximation, unlike Birch's average counting method.