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Finding the roots (zeros) of a given polynomial has been a prominent mathematical problem.. Solving linear, quadratic, cubic and quartic equations in terms of radicals and elementary arithmetic operations on the coefficients can always be done, no matter whether the roots are rational or irrational, real or complex; there are formulas that yield the required solutions.
More specifically, for an ideal I in the ring k[x 1, ..., x n] over a field k, a (Ritt) characteristic set C of I is composed of a set of polynomials in I, which is in triangular shape: polynomials in C have distinct main variables (see the formal definition below). Given a characteristic set C of I, one can decide if a polynomial f is zero ...
One of the easiest examples to check of a Calabi-Yau manifold is given by the Fermat quintic threefold, which is defined by the vanishing locus of the polynomial = + + + + Computing the partial derivatives of gives the four polynomials = = = = = Since the only points where they vanish is given by the coordinate axes in , the vanishing locus is empty since [::::] is not a point in .
Polynomial equations of degree two can be solved with the quadratic formula, which has been known since antiquity. Similarly the cubic formula for degree three, and the quartic formula for degree four, were found during the 16th century. At that time a fundamental problem was whether equations of higher degree could be solved in a similar way.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
Polynomials: Can be generated solely by addition, multiplication, and raising to the power of a positive integer. Constant function: polynomial of degree zero, graph is a horizontal straight line; Linear function: First degree polynomial, graph is a straight line. Quadratic function: Second degree polynomial, graph is a parabola.
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The moduli space of stable curves of genus G is the quotient of a subset of the Hilbert scheme of curves in P 5g–6 with Hilbert polynomial (6n – 1)(g – 1) by the group PGL 5g–5. Example: A vector bundle W over an algebraic curve (or over a Riemann surface) is a stable vector bundle if and only if