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An answer to the P versus NP question would determine whether problems that can be verified in polynomial time can also be solved in polynomial time. If P ≠ NP, which is widely believed, it would mean that there are problems in NP that are harder to compute than to verify: they could not be solved in polynomial time, but the answer could be ...
Algebraic functions are functions that can be expressed as the solution of a polynomial equation with integer coefficients. 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
A polynomial function is one that has the form = + + + + + where n is a non-negative integer that defines the degree of the polynomial. A polynomial with a degree of 0 is simply a constant function; with a degree of 1 is a line; with a degree of 2 is a quadratic; with a degree of 3 is a cubic, and so on.
Every polynomial function is continuous, smooth, and entire. The evaluation of a polynomial is the computation of the corresponding polynomial function; that is, the evaluation consists of substituting a numerical value to each indeterminate and carrying out the indicated multiplications and additions.
Important graph polynomials include: The characteristic polynomial, based on the graph's adjacency matrix. The chromatic polynomial, a polynomial whose values at integer arguments give the number of colorings of the graph with that many colors. The dichromatic polynomial, a 2-variable generalization of the chromatic polynomial
This polynomial is further reduced to = + + which is shown in blue and yields a zero of −5. The final root of the original polynomial may be found by either using the final zero as an initial guess for Newton's method, or by reducing () and solving the linear equation. As can be seen, the expected roots of −8, −5, −3, 2, 3, and 7 were ...
In the mathematical fields of graph theory and combinatorics, a matching polynomial (sometimes called an acyclic polynomial) is a generating function of the numbers of matchings of various sizes in a graph. It is one of several graph polynomials studied in algebraic graph theory.
Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.