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A subvariety of an algebraic variety that is not the entire variety. The proper subvarieties of a line are its nonempty finite subsets, the proper subvarieties of a plane are ...

A metric space Z^^ in which the closure of a congruence class B(j,m) is the corresponding congruence class {x in Z^^|x=j (mod m)}.

The whole neighborhood of any point y_i of an algebraic curve may be uniformly represented by a certain finite number of convergent developments in power series, ...

An algebraic extension K over a field F is a purely inseparable extension if the algebraic number minimal polynomial of any element has only one root, possibly with ...

An extension of an arbitrary field F of the form F(sqrt(1+lambda^2)), where lambda in F.

For a quadratic form Q in the canonical form Q=y_1^2+y_2^2+...+y_p^2-y_(p+1)^2-y_(p+2)^2-...-y_r^2, the rank is the total number r of square terms (both positive and ...

The formula giving the roots of a quadratic equation ax^2+bx+c=0 (1) as x=(-b+/-sqrt(b^2-4ac))/(2a). (2) An alternate form is given by x=(2c)/(-b+/-sqrt(b^2-4ac)). (3)

Given the binary quadratic form ax^2+2bxy+cy^2 (1) with polynomial discriminant b^2-ac, let x = pX+qY (2) y = rX+sY. (3) Then a(pX+qY)^2+2b(pX+qY)(rX+sY)+c(rX+sY)^2 ...

The quantity ps-rq obtained by letting x = pX+qY (1) y = rX+sY (2) in ax^2+2bxy+cy^2 (3) so that A = ap^2+2bpr+cr^2 (4) B = apq+b(ps+qr)+crs (5) C = aq^2+2bqs+cs^2 (6) and ...

A quadratic polynomial is a polynomial of degree 2. A univariate quadratic polynomial has the form f(x)=a_2x^2+a_1x+a_0. An equation involving a quadratic polynomial is ...