In complex analysis, a discipline in mathematics, and in statistical physics, the Asano contraction or Asano–Ruelle contraction is a transformation on a separately affine multivariate polynomial. It was first presented in 1970 by Taro Asano to prove the Lee–Yang theorem in the Heisenberg spin model case. This also yielded a simple proof of the Lee–Yang theorem in the Ising model. David Ruelle proved a general theorem relating the location of the roots of a contracted polynomial to that of the original. Asano contractions have also been used to study polynomials in graph theory.
Let be a polynomial which, when viewed as a function of only one of these variables is an affine function. Such functions are called separately affine. For example, is the general form of a separately affine function in two variables. Any separately affine function can be written in terms of any two of its variables as . The Asano contraction sends to .[1]
Asano contractions are often used in the context of theorems about the location of roots. Asano originally used them because they preserve the property of having no roots when all the variables have magnitude greater than 1.[2] Ruelle provided a more general relationship which allowed the contractions to be used in more applications.[3] He showed that if there are closed sets not containing 0 such that cannot vanish unless for some index , then can only vanish if for some index or where .[4] Ruelle and others have used this theorem to relate the zeroes of the partition function to zeroes of the partition function of its subsystems.