An Investigation into the Roots of Bernstein via Cohomology of Differential Form

Abstract

This work considers the Bernstein polynomial and the methods to calculate it.We consider the polynomial f 2 C[x1;x2; : : : ;xn] and explain how to find the Bernstein polynomial, bf (s), and the operator D 2 An[s], which satisfies D f s+1 = bf (s) f s : The calculation of this polynomial depends on the Weyl algebra, An. We explain these calculations by a lot of examples for one dimensional, two dimensional and three dimensional polynomials. All algorithms which are developed to calculate the Bernstein polynomial did not gave a method to calculate the operator D above, in our method we give a method to how calculate this operator. The main problem of this work is to find a polynomial which describes the dimension of the first cohomology group, ˜b f (s), defined using spaces of one forms and two forms, where f 2 C[x;y]. In more detail, starting with a polynomial f , we define two operators d1 : C!W1 defined by d1(h) = f d(h)+(s+1) d( f ) h; d2 : W1 !W2 defined by d2(w) = f d(w)+s d( f )^w : We then calculate the dimension of first cohomology group for specific values of s. IV The zeros of the polynomial ˜b f (s) is chosen to correspond to the values of s where the cohomology is non-zero. To find a link between our polynomial and the Bernstein polynomial, we compare this with the case when s is the root of the Bernstein polynomial. After a lot of calculation we gave our conjecture that ˜b f (s) is divisible by bf (s). We support our work by a lot of examples when f is a homogeneous polynomial, a quasi-homogeneous polynomial and some more complex cases. The original motivation for our problem is to study the first integrals of vector fields with Darboux integrating factor in the case where the first integral is not of Darboux form. The roots of ˜b f (s) are exactly the cases where we find new classes of integrals.