noun
definition
A bihomogeneous polynomial in x, y, ... and the coefficients of some homogeneous form in x, y, ... that is invariant under some group of linear transformations.
definition
The variety defined by a covariant.
adjective
definition
(Of a functor) which preserves composition.
definition
Using or relating to covariance.
Examples of covariant in a Sentence
If the forms be identical the sets of symbols are ultimately equated, and the form, provided it does not vanish, is a covariant of the form ate.
For the cubic (ab) 2 axbx is a covariant because each symbol a, b occurs three times; we can first of all find its real expression as a simultaneous covariant of two cubics, and then, by supposing the two cubics to merge into identity, find the expression of the quadratic covariant, of the single cubic, commonly known as the Hessian.
It will be apparent that there are four numbers associated with a covariant, viz.
The two forms ax, bx, or of, 0, may be identical; we then have the kth transvectant of a form over itself which may, or may not, vanish identically; and, in the latter case, is a covariant of the single form.
For the quadratic it is the discriminant (ab) 2 and for ax2 the cubic the quadratic covariant (ab) 2 axbx.
In particular, when the product denotes an invariant we may transform each of the symbols a, b,...to x in succession, and take the sum of the resultant products; we thus obtain a covariant which is called the first evectant of the original invariant.
Similarly regarding 1 x 2 as additional parameters, we see that every covariant is expressible as a rational function of n fixed covariants.
Every covariant is rationally expressible by means of the forms f, u 2, u3,...
To exhibit any covariant as a function of uo, ul, a n = (aiy1+a2y2) n and transform it by the substitution fi y 1+f2 y where f l = aay 1, f2 = a2ay -1, x y - x y = X x thence f .
We will write the cubic covariant (f, i) 2 =j, and then remark that the result, (f,j) 3 = o, can be readily established.
The form j is completely defined by the relation (f,j) 3 =o as no other covariant possesses this property.
On this principle the covariant j is expressible in the form R 2 j =5 3 + BS 2 a+4ACSa 2 + C(3AB -4C)a3 when S, a are the above defined linear forms.
Now, evidently, the third transvectant of f, expressed in this form, with the cubic pxgxrx is zero, and hence from a property of the covariant j we must have j = pxgxrx; showing that the linear forms involved are the linear factors of j.
Such a symbolic product, if its does not vanish identically, denotes an invariant or a covariant, according as factors az, bz, cz,...
We can see that (abc)a x b x c x is not a covariant, because it vanishes identically, the interchange of a and b changing its sign instead of leaving it unchanged; but (abc) 2 is an invariant.
One more covariant is requisite to make an algebraically complete set.
When R =0, and neither of the expressions AC - B 2, 2AB -3C vanishes, the covariant a x is a linear factor of f; but, when R =AC - B 2 = 2AB -3C =0, a x also vanishes, and then f is the product of the form jx and of the Hessian of jx.
It has been shown above that a covariant, in general, satisfies four partial differential equations.
Two of these show that the leading coefficient of any covariant is an isobaric and homogeneous function of the coefficients of the form; the remaining two may be regarded as operators which cause the vanishing of the covariant.
It is for this reason called a seminvariant, and every seminvariant is the leading coefficient of a covariant.
If now the nti c denote a given pencil of lines, an invariant is the criterion of the pencil possessing some particular property which is independent alike of the axes and of the multiples, and a covariant expresses that the pencil of lines which it denotes is a fixed pencil whatever be the axes or the multiples.
The general form of covariant is therefore (ab) h i (ac) h2 c) (b h3 a i bb2c'e3...abia?2b?3...
It may denote a simultaneous orthogonal invariant of forms of orders n i, n2, n3,...; degree 0 of the covariant in the coefficients.
Hence in the above general form of covariant we may suppose the exponents h 1, h2, h3,...ki, k2, k3,...
Every other concomitant is a rational integral function of these four forms. The linear covariant, obviously the Jacobian of a x and x x is the line perpendicular to x and the vanishing of the quadrinvariant a x is the condition that a x passes through one of the circular points at infinity.
There is no linear covariant, since it is impossible to form a symbolic product which will contain x once and at the same time appertain to a quadratic. (v.) is the Jacobian; geometrically it denotes the bisectors of the angles between the lines ax, or, as we may say, the common harmonic conjugates of the lines and the lines x x .
We have in the Hessian the first instance of a covariant of a ternary form.
The Hessian A has just been spoken of as a covariant of the form u; the notion of invariants and covariants belongs rather to the form u than to the curve u=o represented by means of this form; and the theory may be very briefly referred to.
Similarly, if we have a curve U= o derived from the curve u = o in a manner independent of the particular axes of co-ordinates, then from the transformed equation u' = o deriving in like manner the curve U' = o, the two equations U= o, U' = o must each of them imply the other; and when this is so, U will be a covariant of u.
The case is less frequent, but it may arise, that there are covariant systems U= o, V = o, &c., and U' = o, V' = o, &c., each implying the other, but where the functions U, V, &c., are not of necessity covariants of u.
By simple multiplication (al b l b2 -24a2bib2+ala2b;)xi +(aibz -ala214b2-aia2blb2+a2b2)xlx2 + (aia 2 b2 - 2a l a2b l b2 +a2/4b 2)x2; and transforming to the real form, (aob 2 - 2a1b,+a2bo)xi (aob 3 -a l b 2 - alb,+a3bo)xlx2 + (aib3 - 2a2b2+a3b1)x2, the simultaneous covariant; and now, putting b = a, we obtain twice.
From these formulae we derive two important relations, dp4 = or the function F, on the right which multiplies r, is said to be a simultaneous invariant or covariant of the system of quantics.
It is always an invariant or covariant appertaining to a number of different linear forms, and as before it may vanish if two such linear forms be identical.
In general it will be simultaneous covariant of the different forms n 1 rz 2 n3 a, b x, ?