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2. Hilbert's seventeenth problem - Wikipedia

   en.wikipedia.org/wiki/Hilbert's_seventeenth_problem

   Hilbert's proof did not exhibit any explicit counterexample: only in 1967 the first explicit counterexample was constructed by Motzkin. [3] Furthermore, if the polynomial has a degree 2d greater than two, there are significantly many more non-negative polynomials that cannot be expressed as sums of squares. [4]

3. Degree of a polynomial - Wikipedia

   en.wikipedia.org/wiki/Degree_of_a_polynomial

   For polynomials in two or more variables, the degree of a term is the sum of the exponents of the variables in the term; the degree (sometimes called the total degree) of the polynomial is again the maximum of the degrees of all terms in the polynomial. For example, the polynomial x 2 y 2 + 3x 3 + 4y has degree 4, the same degree as the term x ...

4. Horner's method - Wikipedia

   en.wikipedia.org/wiki/Horner's_method

   The largest zero of this polynomial which corresponds to the second largest zero of the original polynomial is found at 3 and is circled in red. The degree 5 polynomial is now divided by () to obtain = + + which is shown in yellow. The zero for this polynomial is found at 2 again using Newton's method and is circled in yellow.

5. Abel–Ruffini theorem - Wikipedia

   en.wikipedia.org/wiki/Abel–Ruffini_theorem

   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.

6. Polynomial - Wikipedia

   en.wikipedia.org/wiki/Polynomial

   In the case of polynomials in more than one indeterminate, a polynomial is called homogeneous of degree n if all of its non-zero terms have degree n. The zero polynomial is homogeneous, and, as a homogeneous polynomial, its degree is undefined. [c] For example, x 3 y 2 + 7x 2 y 3 − 3x 5 is homogeneous of degree 5.

7. Quintic function - Wikipedia

   en.wikipedia.org/wiki/Quintic_function

   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

8. Polynomial greatest common divisor - Wikipedia

   en.wikipedia.org/wiki/Polynomial_greatest_common...

   Its existence is based on the following theorem: Given two univariate polynomials a and b ≠ 0 defined over a field, there exist two polynomials q (the quotient) and r (the remainder) which satisfy = + and ⁡ < ⁡ (), where "deg(...)" denotes the degree and the degree of the zero polynomial is defined as being negative.

9. Multilinear polynomial - Wikipedia

   en.wikipedia.org/wiki/Multilinear_polynomial

   The resulting polynomial is not a linear function of the coordinates (its degree can be higher than 1), but it is a linear function of the fitted data values. The determinant, permanent and other immanants of a matrix are homogeneous multilinear polynomials in the elements of the matrix (and also multilinear forms in the rows or columns).