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Therefore, let f(x) = g(x) = 2x + 1. Then, f(x)g(x) = 4x 2 + 4x + 1 = 1. Thus deg(f⋅g) = 0 which is not greater than the degrees of f and g (which each had degree 1). Since the norm function is not defined for the zero element of the ring, we consider the degree of the polynomial f(x) = 0 to also be undefined so that it follows the rules of a ...
With modern computers and programs, deciding whether a polynomial is solvable by radicals can be done for polynomials of degree greater than 100. [6] Computing the solutions in radicals of solvable polynomials requires huge computations. Even for the degree five, the expression of the solutions is so huge that it has no practical interest.
Knot at infinity of X 0 (11) The classical modular curve, which we will call X 0 (n), is of degree greater than or equal to 2n when n > 1, with equality if and only if n is a prime. The polynomial Φ n has integer coefficients, and hence is defined over every field. However, the coefficients are sufficiently large that computational work with ...
If a and b are rational numbers, the equation x 5 + ax + b = 0 is solvable by radicals if either its left-hand side is a product of polynomials of degree less than 5 with rational coefficients or there exist two rational numbers ℓ and m such that
Rather, the degree of the zero polynomial is either left explicitly undefined, or defined as negative (either −1 or −∞). [10] The zero polynomial is also unique in that it is the only polynomial in one indeterminate that has an infinite number of roots. The graph of the zero polynomial, f(x) = 0, is the x-axis.
Next consider the values of polynomials at a complex number x, when these polynomials have integer coefficients, degree at most n, and height at most H, with n, H being positive integers. Let (,,) be the minimum non-zero absolute value such polynomials take at and take:
In this case, the product of the degrees of the polynomials may be much larger than the number of roots, and better bounds are useful. Multi-homogeneous Bézout theorem provides such a better bound when the unknowns may be split into several subsets such that the degree of each polynomial in each subset is lower than the total degree of the ...
W(x) has no zeros or infinities inside the interval, though it may have zeros or infinities at the end points. W(x) gives a finite inner product to any polynomials. W(x) can be made to be greater than 0 in the interval. (Negate the entire differential equation if necessary so that Q(x) > 0 inside the interval.)