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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.

33ALGEBRAIC 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.

33For 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=ï¿½.., Jz=..., and it appears that the vanishing of u, v, and w implies the vanishing of J.

23CY 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.

23Case of Three Variables.-In the next place we consider the resultants of three homogeneous polynomials in three variables.

24Under 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.

14Discriminants.-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.

14It 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).

14Xic-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 = ...

14Under 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.

14It is practically identical with that of finding the greatest common measure of two polynomials.

00Lower order polynomials are trivial to solve while higher order polynomials require iterative algorithms to solve them.

00chromatic polynomials of families of graphs.

00copula function using Bernstein polynomials is studied.

00Thus we don't ever need to compute the Bezier polynomials, we simply depth reduce the control points recursively until d =0.

00These methods are based on a discontinuous Galerkin approach, where the unknowns are approximated by completely discontinuous piecewise polynomials.

00divisor of these polynomials.

00To calculate the greatest common divisor of two integers and of two polynomials over a field.

00multivariate polynomials in order to derive integers makes no sense!

00orthogonal polynomials play an important role in the analysis.

00Here, orthogonal polynomials play an important role in the analysis.

00Reminder: How to find the roots of quadratic polynomials (pdf ).

00I shall discuss recent developments in the ' transfer matrix ' method for calculating chromatic polynomials of families of graphs.

00Clearly writing a class hierarchy starting with multivariate polynomials in order to derive integers makes no sense!

00GAP Forum: Re: a problem with factoring polynomials Subject: Re: a problem with factoring polynomials.

00quadratic polynomials (pdf ).

00Examples of polynomials which are not solvable by radicals.

00Case of Three Variables.-In the next place we consider the resultants of three homogeneous polynomials in three variables.

00For 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=Ã¯¿½.., Jz=..., and it appears that the vanishing of u, v, and w implies the vanishing of J.

00CY 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.

00Discriminants.-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.

00It 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).

00Xic-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 = ...

00It is practically identical with that of finding the greatest common measure of two polynomials.

00Examples of polynomials which are not solvable by radicals.

00

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