Kac-Rice formulas for random fields and theirs applications in: random geometry, roots of random polynomials and some engineering problems

International audience ; There exist two variants of the change of variables formula for multiple integrals very useful in integral geometry.The first one corresponds to smooth, locally bijective functions G from Rd to Rd and the second applies to smooth functions G from Rd to Rj with d>j, having a differential with maximal rank.These formulas are called "area formula" and "coarea formula" respectively.Applying these formulas to trajectories of random fields and taking expectation afterwards, one obtains the well-known Kac-Rice formulas.In recent times and fundamentally due to the appearance of two excellent books (Adler and Taylor, 2007) and (Azaïs and Wschebor, 2009), there has been a growing interest in the application of these formulas in such varied domains as: random algebraic geometry, algorithm complexity for solving large systems of equations, study of zeros of random polynomial systems and finally, engineering applications.The present work is divided in three parts.1. In the first part, we give an analytical proof of the area and coarea formulas. Such a proof, originally attributed to Banach and Federer (1969), will be made by using elementary tools of vector calculus and measure theory in Rd. 2. The above formulas form the basis for establishing the validity of Kac-Rice formulas for random fields. They allow computing the expectation of the measure of the level sets C_{Q,X}(y)={t in Q \subset Rd: X(t)=y}, where X: Omega x Rd to Rj is a random fields and d \ge j. We must point out that one can obtain a Kac-Rice formula for almost sure all level by using the area and coarea formulas, Fubini theorem and duality. However, in applications the interest is directed to a fixed level y. For instance, the zeros in the study of the roots of a random polynomial. This precision leads us to a delicate study for generalizing the classical inverse function and implicit function theorems. For this part we based our approach in two seminal works: firstly an article of E.Cabana (1985), published in the conference ...