What are the mathematical problems of variables?

The following is the mathematical problem of invariants.

1, the invariant of rigid body transformation group-Euclidean geometry invariant

Under the mixed action of rotation and translation, the geometric invariance is constant, and there is distance, area and volume here. Self-evident.

2. Invariant under * * shape transformation group-Euclidean geometry invariant

Rotation, translation and scaling generate a larger group than the above groups. It no longer maintains its length, area and volume. Since Qiu Chengtong pioneered computational geometry, this group has become more and more important.

People who do artificial intelligence algorithms will tell you that this set of invariants that act on people's faces are called face features, what are face features-whether three courts and five eyes are even, and other information that any fortune teller needs. Point * * * Point of line * *, roundness of four points, included angle of three points, intersection ratio of points of line * * *, proportion of line segments and so on are all invariants under the action of this group.

Looking back at the transformation of quadratic form f(x, y, z)=0, the inertia index (the ultimate invariant of quadratic form) will be transformed by * * * shape group.

3. Invariants under projective transformation groups-projective geometric invariants.

Groups generated by rotation, translation, scaling and Mobius transformation. Its characteristic is that a cubic and quartic equation with a homogeneous f(x, y, z)=0 with ten coefficients (using quartic Mobius transformation) is transformed into the form of y 2 = x 3+ax+b, and the final J invariant can be obtained.

Among them, the geometric meaning of Mobius transformation is to project a spatial curve from a point to a plane. This is also the reason why quartic curve can be cast as Wilstrass standard on XOY plane. Because the losses are all 1.

The concept of modular space with genus 1 is introduced. Obviously, the dimension of the module space of genus 1 is 1 under projective equivalence. The invariant is 1 (dimension), which gives us the courage and motivation to constantly toss about various permutations, because we believe that we can finally exchange an equation with only one unknown coefficient.

4. Double rational invariants-invariants of projective geometry

Birational transformation pairs X = f (x, y) and Y = g (x, y) are also birational at first, so what is its transformation group? Can bi-rational transformation be decomposed into basic singularity elimination transformation? X = xyY = y Continuing to ask this question may irritate Xu Chenyang.

Japanese algebraic geometer Hironaka Youping has done a basic work. He solved all the singularities defined by plane polynomial curves. This means that some birational transformations are generated by the transformation of singularity elimination.