We present formulas for the multivariate resultant as a quotient of two determinants. They extend the classical Macaulay formulas, and involve matrices of considerably smaller size, whose non zero entries include coefficients of the given polynomials and coefficients of their Bezoutian. These formulas can also be viewed as an explicit computation of the morphisms and the determinant of a result...

This is the 4-th paper in the series devoted to a systematic study of the problem of mathematically correct formulation of the rules needed to manage an effective field theory. Here we consider the problem of constructing the full set of essential parameters in the case of the most general effective scattering theory containing no massless particles with spin J > 1/2. We perform the detailed cl...

The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of n homogeneous equations in n variables is satisfiable (the resultant is a polynomial in the system’s coefficients which vanishes if and only if the system is satisfiable). In this paper, we investigate the complexity of computing the multivariate resul...

Electronic cigarettes (e-cigarettes) are growing in popularity exponentially. Despite their ever-growing acceptance, their aerosol has not been fully characterized. The current study focused on evaluating e-cigarette solutions and their resultant aerosol for potential differences. A simple sampling device was developed to draw e-cigarette aerosol into a multi-sorbent thermal desorption (TD) tub...

Background: Surgical skin incisions in rhytidoplasty are very diverse compared to their original form approximately 100 years ago, when the operation was first performed. Many strategies have been proposed to improve this procedure so as to reduce scarring, anatomic distortion of the external ear, and hair loss. To diminish the inconveniences of periauricu lar scars and hair loss and in order t...

Remark 1.2 The Fundamental Theorem of Algebra enters in the very last inequality: the fact that having d distinct roots implies that Res(d,d−1)(f, f ) 6= 0. Later we will see a refinement of the above theorem giving a positive lower bound even when Res(d,d−1)(f, f )=0. ⋄ A key result we’ll need is the following algebraic identity. Lemma 1.3 Following the notation of Theorem 1.1, we have Res(d,d...