ALGEBRAIC FORMS. The subject-matter of algebraic forms is to a large extent connected with the linear transformation of algebraical polynomials which involve two or more variables.

ALGEBRAIC FORMS. The subject-matter of algebraic forms is to a large extent connected with the linear transformation of algebraical polynomials which involve two or more variables.

For if u, v, w be the polynomials of orders m, n, p respectively, the Jacobian is (u 1 v 2 w3), and by Euler's theorem of homogeneous functions xu i +yu 2 +zu 3 = mu xv1 +yv2 +zv3 = /IV xw 1+y w 2+ zw 3 = pw; denoting now the reciprocal determinant by (U 1 V2 W3) we obtain Jx =muUi+nvVi+pwWi; Jy=�.., Jz=..., and it appears that the vanishing of u, v, and w implies the vanishing of J.

CY The proof being of general application we may state that a system of values which causes the vanishing of k polynomials in k variables causes also the vanishing of the Jacobian, and in particular, when the forms are of the same degree, the vanishing also of the differential coefficients of the Jacobian in regard to each of the variables.

Case of Three Variables.-In the next place we consider the resultants of three homogeneous polynomials in three variables.

Under the general heading "Algebra and Theory of Numbers" occur the subheadings "Elements of Algebra," with the topics rational polynomials, permutations, &c., partitions, probabilities; "Linear Substitutions," with the topics determinants, &c., linear substitutions, general theory of quantics; "Theory of Algebraic Equations," with the topics existence of roots, separation of and approximation to, theory of Galois, &c. "Theory of Numbers," with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers.

Discriminants.-The discriminant of a homogeneous polynomial in k variables is the resultant of the k polynomials formed by differentiations in regard to each of the variables.

It is the resultant of k polynomials each of degree m-I, and thus contains the coefficients of each form to the degree (m-I)'-1; hence the total degrees in the coefficients of the k forms is, by addition, k (m - 1) k - 1; it may further be shown that the weight of each term of the resultant is constant and equal to m(m-I) - (Salmon, l.c. p. loo).

Xic-1, the coefficients being any polynomials, it is clear that the k differentials have, in common, the system of roots derived from X1= X 2 = ...

Under the general heading "Algebra and Theory of Numbers" occur the subheadings "Elements of Algebra," with the topics rational polynomials, permutations, &c., partitions, probabilities; "Linear Substitutions," with the topics determinants, &c., linear substitutions, general theory of quantics; "Theory of Algebraic Equations," with the topics existence of roots, separation of and approximation to, theory of Galois, &c. "Theory of Numbers," with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers.

It is practically identical with that of finding the greatest common measure of two polynomials.

Lower order polynomials are trivial to solve while higher order polynomials require iterative algorithms to solve them.

chromatic polynomials of families of graphs.

copula function using Bernstein polynomials is studied.

Thus we don't ever need to compute the Bezier polynomials, we simply depth reduce the control points recursively until d =0.

These methods are based on a discontinuous Galerkin approach, where the unknowns are approximated by completely discontinuous piecewise polynomials.

divisor of these polynomials.

To calculate the greatest common divisor of two integers and of two polynomials over a field.

multivariate polynomials in order to derive integers makes no sense!

orthogonal polynomials play an important role in the analysis.

Here, orthogonal polynomials play an important role in the analysis.

Reminder: How to find the roots of quadratic polynomials (pdf ).

I shall discuss recent developments in the ' transfer matrix ' method for calculating chromatic polynomials of families of graphs.

Clearly writing a class hierarchy starting with multivariate polynomials in order to derive integers makes no sense!

GAP Forum: Re: a problem with factoring polynomials Subject: Re: a problem with factoring polynomials.

quadratic polynomials (pdf ).

Examples of polynomials which are not solvable by radicals.

Case of Three Variables.-In the next place we consider the resultants of three homogeneous polynomials in three variables.

For if u, v, w be the polynomials of orders m, n, p respectively, the Jacobian is (u 1 v 2 w3), and by Euler's theorem of homogeneous functions xu i +yu 2 +zu 3 = mu xv1 +yv2 +zv3 = /IV xw 1+y w 2+ zw 3 = pw; denoting now the reciprocal determinant by (U 1 V2 W3) we obtain Jx =muUi+nvVi+pwWi; Jy=�.., Jz=..., and it appears that the vanishing of u, v, and w implies the vanishing of J.

CY The proof being of general application we may state that a system of values which causes the vanishing of k polynomials in k variables causes also the vanishing of the Jacobian, and in particular, when the forms are of the same degree, the vanishing also of the differential coefficients of the Jacobian in regard to each of the variables.

Discriminants.-The discriminant of a homogeneous polynomial in k variables is the resultant of the k polynomials formed by differentiations in regard to each of the variables.

It is the resultant of k polynomials each of degree m-I, and thus contains the coefficients of each form to the degree (m-I)'-1; hence the total degrees in the coefficients of the k forms is, by addition, k (m - 1) k - 1; it may further be shown that the weight of each term of the resultant is constant and equal to m(m-I) - (Salmon, l.c. p. loo).

Xic-1, the coefficients being any polynomials, it is clear that the k differentials have, in common, the system of roots derived from X1= X 2 = ...

It is practically identical with that of finding the greatest common measure of two polynomials.

Examples of polynomials which are not solvable by radicals.