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.

Date of Award | 2021 |
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Original language | English |
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Awarding Institution | |
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Supervisor | Colin Christopher (Other Supervisor) |
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An Investigation into the Roots of Bernstein via Cohomology of Differential Form

Noor, A. (Author). 2021

Student thesis: PhD