Ueber einen Satz aus der Theorie der algebraischen Functionen

Residue currents of holomorphic morphisms

“…Therefore there is a solution to P Q = Φ such that deg P j Q j ≤ deg Φ. When r = 1 this is a classical theorem due to Max Noether, [18]. We have the following generalization that appeared in [5] in the case r = 1; however we suspect that this case could be proved algebraically, e.g., by the methods in [15].…”

Sparse effective membership problems via residue currents

“…Theorem 1.2 also holds if P is of the form (1.7). In particular, if m = n and P = (deg Φ)Σ n we get back Nöther's original result [27]: Assume that the zero-set of F 1 , . .…”

“…Observe that supp F ⊆ P, where P is given by (1.7), means that the degree in the first n 1 variables are bounded by d 1 , the degree in the next n 2 variables are bounded by d 2 , etc. Our next result is a sparse version of Max Nöther's AF + BG Theorem, [27]. Theorem 1.2.…”

“…In many cases, however, one can do much better. Classical results due to Max Nöther [27] and Macaulay [26] show that the bounds can be substantially improved if (the homogenizations of) the F j have no zeros at infinity.…”

Geometry at Cambridge, 1863–1940

“…By the middle of the 19th century, it was widely believed that any curve through these points had an equation of the form AF + BG = 0, where the equations of the curves C m and C n are F = 0 and G = 0, respectively, and A and B are also polynomials in x and y. In an influential paper Noether [1873] had shown that this belief was false, notably when the given curves were tangent to each other or had singular points. The next year Brill and Noether [1874] had dealt with the question at length, and established the fundamental theorem in the subject, the Riemann-Roch theorem, which they were the first to call by this name.…”