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Given a homogeneous polynomial of degree with real coefficients that takes only positive values, one gets a positively homogeneous function of degree / by raising it to the power /. So for example, the following function is positively homogeneous of degree 1 but not homogeneous: ( x 2 + y 2 + z 2 ) 1 2 . {\displaystyle \left(x^{2}+y^{2}+z^{2 ...
A polynomial of degree 0 is always homogeneous; it is simply an element of the field or ring of the coefficients, usually called a constant or a scalar. A form of degree 1 is a linear form. [notes 2] A form of degree 2 is a quadratic form. In geometry, the Euclidean distance is the square root of a quadratic form.
Let H be the homogeneous ideal generated by the homogeneous parts of highest degree of the elements of I. If I is homogeneous, then H=I. Finally let B be a Gröbner basis of I for a monomial ordering refining the total degree partial ordering and G the (homogeneous) ideal generated by the leading monomials of the elements of B.
The resultant of two polynomials depending on a variable x and other variables is a polynomial in the other variables that is in the ideal generated by the two polynomials, and has the following properties: if one of the polynomials is monic in x, every zero (in the other variables) of the resultant may be extended into a common zero of the two ...
The multi-homogeneous Bézout bound on the number of solutions may be used for non-homogeneous systems of equations, when the polynomials may be (multi)-homogenized without increasing the total degree. However, in this case, the bound may be not sharp, if there are solutions "at infinity".
In other words, every homogeneous polynomial of degree d with more than d 2 variables has a non-trivial zero (so F p ((t)) is a C 2 field). Then one shows that if two Henselian valued fields have equivalent valuation groups and residue fields, and the residue fields have characteristic 0, then they are elementarily equivalent (which means that ...
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] The following table summarizes in which cases every non-negative homogeneous polynomial (or a polynomial of even degree) can be represented as a sum of squares:
A linear differential equation is homogeneous if it is a homogeneous linear equation in the unknown function and its derivatives. It follows that, if φ(x) is a solution, so is cφ(x), for any (non-zero) constant c. In order for this condition to hold, each nonzero term of the linear differential equation must depend on the unknown function or ...