note-math

In flat space, linear and affine are often mixed. Similarly, in flat space, so are polynomials. Zero-order polynomials correspond to the use of affine

First deal with the case of one-dimensional real numbers

exponential power function

[polynomial-function-1d] A polynomial function is a finite linear combination of power functions. (Affine) base point , (vector) offset

Polynomial function representation is not affine invariant, i.e. switching the base point will result in a polynomial function representation of the same order but with different coefficients. Scaling is also the case

[change-base-point-polynomial] Switch base point

Represents the same affine function

then

Proof expand the calculation, to compare coefficients, collect power function terms, by the exchange of summation

If the base point is in the coordinates and the symbol is changed, then the polynomial function is expressed as

Extending from polynomial as a finite linear combination to a countably infinite linear combination is called the exponential power series of a function

The definitions of some functions do not come directly from the exponential power series, Example

In addition to as countably infinite data, can also be used. The exponential power function is changed to the exponential power function

• requires multiplicative inverse

• requires solving the equation and needs to deal with the issue of whether the number of solutions is unique

• is unbounded at

• When , the multiple derivatives will not be interrupted

Here we only deal with power series for the time being, and refer to them as power series for short

Now deal with the high-dimensional case i.e.

If the range is , we can also define the multiplication of functions and the multiplicative inverse

First try to define polynomial functions and power series based on tensors i.e. multilinear

If not necessary, there is no need to take the linear direct sum of tensors of all orders (called tensor algebra)

[polynomial-function] Using the range and multilinear function . Base point , offset , define polynomial function

Affine transformations, i.e., changing the base point i.e. translation, or linear transformations i.e. (including scaling), do not change the order of the polynomial

may not be collinear

Extending to is simple. For the cases of , due to non-commutativity and non-associativity, high-dimensional linear algebra and tensors require new processing methods

Different tensors may give the same polynomial, but the symmetric tensor is uniquely corresponding

Change notation

• [power-tensor]

The method to restore from a monomial of order or a power tensor to a symmetric tensor of order is difference

In the -th order monomial of , there is a term , but there are many other interfering terms

The whole problem is symmetric with respect to , so a symmetric construction should be used

In the second order

[difference-symmetric-tensor] Symmetric tensor -th order difference

Question Is there an intuitive understanding of the -th order difference?

[successive-difference] The -th order difference can be written as times the first-order difference

where , ,

Due to the commutativity of summation, the order of successive differences does not affect the final result

Proof of difference-symmetric-tensor

The symmetry of the symmetric multilinear function allows the properties of difference to be inherited

[difference-polynomial] The -th order difference of is

The -th order difference of is

From this, we get that the polynomial function determines its multiple symmetric linear function representation Proof First, -difference gives the same , after removing from both sides, it is still the same polynomial function, the order is , continue difference to get the same …

For power series, finite-order difference can never give zero

Formally, division and limits can be used to eliminate higher-order terms