The tangential Hilbert 16th problem is to place an upper bound for the number of isolated ovals of algebraic level curves {H(x, y) = const} over which the integral of a polynomial 1-form P (x, y) dx + Q(x, y) dy (the Abelian integral) may vanish, the answer to be given in terms of the degrees n = degH and d = max(degP, degQ). We describe an algorithm producing this upper bound in the form of a ...

We consider the number of zeros of the integral I(h) = ∮ Γh ω of real polynomial form ω of degree not greater than n over a family of vanishing cycles on curves Γh : y 2 + 3x − x = h, where the integral is considered as a function of the parameter h. We prove that the number of zeros of I(h), for 0 < h < 2, is bounded above by 2[n−1 2 ] + 1.

where f (x, y) and g(x, y) are real polynomials of x and y with degree not greater than n, 1h is an oval lying on real algebraic curve H(x, y)=h, deg H(x, y)=m (H(x, y) are called Hamiltonians), and 7 is a maximal interval of existence of 1h . This question is called the weakened Hilbert 16th problem, posed by V.I. Arnold in [1, 2]. The general result of solving the weakened Hilbert 16th proble...

In this paper, we study the Chebyshev property of the 3-dimentional vector space $E =langle I_0, I_1, I_2rangle$, where $I_k(h)=int_{H=h}x^ky,dx$ and $H(x,y)=frac{1}{2}y^2+frac{1}{2}(e^{-2x}+1)-e^{-x}$ is a non-algebraic Hamiltonian function. Our main result asserts that $E$ is an extended complete Chebyshev space for $hin(0,frac{1}{2})$. To this end, we use the criterion and tools developed by...

A new procedure for regularizing Feynman integrals in the noncovariant Coulomb gauge ~ ∇· ~ A = 0 is proposed for Yang-Mills theory. The procedure is based on a variant of dimensional regularization, called split dimensional regularization, which leads to internally consistent, ambiguity-free integrals. It is demonstrated that split dimensional regularization yields a one-loop Yang-Mills self-e...