Mathematics
The Definition of Polynomial Jacobian
Polynomial Jacobian {*} should refer to the process of generating the image of polynomials. This process involves a calculation framework for assigning coefficients based on the existence of a solution set (xi). The result is an equation system that provides an equation for each coefficient in terms of the solution set. One should understand this process as if, locally, there is a real function that allows a series of values to return a zero output. Therefore, it is similar to the definition of a manifold {1} or algebraic variety {2} in the sense of its already explored characteristics. However, we should understand this process as a pure model in which there is a familiar expression to describe its mathematics. Because of its nature, coefficients are image to the solution set, for their combined individual manifold holds the algebraic varieties to be true, to be null locally at the solution set. There are as many equations for the coefficients as there are solutions in the solution sets. Thus, this calculation framework usefulness extends beyond generating coefficients, for it should be used for examining the equation system with regards to the coefficients.
{*}:= Polynomial Jacobian. The Definition of Polynomial Jacobian. Its characteristics are going to be described in this essay. I have chosen this interesting expression because of its current definitions share similarities of concepts with the previous key ideas such as manifold, square matrix, null-vector, etc. Meantime, there is none direct involvement of those early works throughout the definition of Polynomial Jacobian.
{1}:= Manifold. (2022, March 28). In Wikipedia. {“In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.”}
{2}:= Algebraic variety. (2022, March 22). In Wikipedia. {“Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers.”}
Since all coefficients are proportional to the inverse product of the solution set, we will admit upon the manifold that the inverse of the new functions for the coefficients exist over the real numbers. Because of the solution sets exclude the value zero, that is making valid the reference to an algebraic variety, from a manifold, that is being derived from the classic polynomial equation p(x) = 0. Indeed, analyzing polynomials with knowledge of Polynomial Jacobian is a mean to determine as whether or not a given series of coefficients should be valid. Because of locally to the solution set, the global function is null, belongs to the real numbers, and its acting as the inverse of the global function exists conversely to the definition of the coefficients, for any order of magnitude.
One may have already in mind, with regards to familiarity of polynomials, that the roots of unity {3} must be defined, as well, in the broader context of the Polynomial Jacobian. Despite the historic sentiment of accomplishment towards Contemporary results, still the roots of unity are contended in a narrow window of the Polynomial Jacobian, in which the equation system admits trivial values, such as observing both non-zero and zero values being assigned to the coefficients. Consequently, either the notion of symmetry or non-symmetry among the relationships in the solution set does not complement our understanding of the Polynomial Jacobian. Instead, it is an underlying observation from the scrutiny of the algebraic varieties. Meantime, the notion of manifold suggests us that the assumed invertible mapping property, to be respected upon a condition of complex variables, would require those coefficients to be null to undoing the complex functions locally.
{3}:= Root of unity. (2022, March 16). In Wikipedia. https://en.wikipedia.org/wiki/Root_of_unity . {“In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power n.”}
We have considered more useful to count the coefficient letters from A towards Z though it emphasizes the reliability of solutions between the range ]0, 1]. In which case, the order of related magnitude to the variable x, in the global function p(x) = 0, does not imply that this term contributes more to the output.
In addition, we have assumed that ambiguity arises from the classic writing of the global function p(x) = 0, when numerous authors are conveying the idea that the global function is analytic in terms of p(x) - c = c. Because of this habit implies that the coefficient with the highest order of magnitude is bound to take the neutral value. We would rather prefer that any constant value remains unscaled. However, one should agree with us that the equation system is allowing a trivial value when we are considering the constant term. As a matter of consistency in the equation system, we are defining (n) equations only. We believe in the benefit of this proposal, for the Polynomial Jacobian should express a more intuitive syntax in carrying the new analytic tool for polynomials, in the Cartesian plane.
Furthermore, we have noticed that community scientists would be preferring to view the polynomial form of the exponential {4}, in terms of coefficients such as 1/(n!). Although, we firmly discourage readers to blending their understanding of the two expressions together, that is for the polynomial form of the exponential and the Polynomial Jacobian. Because of there is none algebraic variety equivalency between the two concepts of coefficient series.
{4}:= Characterizations of the exponential function. (2022, March 29). In Wikipedia.
https://en.wikipedia.org/wiki/Characterizations_of_the_exponential_function . {“In mathematics, the exponential function can be characterized in many ways.”}
Moreover, there is a need to view the definition of Polynomial Jacobian as if it were a vectoral {5} approach instead of a geometrical approach towards synthesizing the global function. Because of the solution set is implying the existence of the global function, while the effect of invertible mapping in the materials arrangement of the coefficients really is analytical. For example, the vector p(x):= [A, B, C] would refer to p(X):= [f(X1,X2), g(X1,X2), 1]. In this context, it is representing the concept of manifold and it enables invertible mapping upon the global function as if it is behaving like a zero vector. One may agree with us that the global function has an only independent variable. Herein, we want to describe uni-dimensional curves which their influx is going to vary with respect to a sum of combined contributions from the coefficient and variable terms.
{5}:= Null vector. (2022, March 29). In Wikipedia.
https://en.wikipedia.org/wiki/Null_vector . {“In mathematics, given a vector space X with an associated quadratic form q, written (X, q), a null vector or isotropic vector is a non-zero element x of X for which q(x) = 0.”}
This article aims at laying on paper the aspects of the calculation framework. Given the definition of polynomials, p(x) = A + Bx + Cx² + Dx³ + Ex⁴…, solving for the variable (x) in the equation requires to assume p(x) = 0, when it is evaluated at a specific solution set (xi), x1, x2, x3, x4…. The definition of the Polynomial Jacobian is an expression of the coefficients:
J(p(x)) = ∑ (-1)^ni*Ω{∏xi}ni / ∏(xi);
And, it expresses conditions upon the coefficients based on the solution set. It means “Coefficients are equal to the sum of set products xi (with regards to its related order of magnitude), bares a minus sign only if the order of magnitude is odd, and they are divided by the product of all xi.” One may recognize, in the choice for a typology of formulated symbols, the symbols for a set of events, but it implies that the solution set is being expressed in the form of deducted products, which they are complementary to one another. For example, the sum of terms 1/x1 + 1/x2 + 1/x3 + 1/x4; so, it should be rewritten as a sum of complementary products of the solution set
Ω (xi) = (x1, x2, x3, x4): Ω{∏xi}/ ∏(xi) =
(x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4)/(x1*x2*x3*x4).
Hereby, one recognizes that the larger the solution set, the higher the number of terms there are in the sum of complementary products of the solution set. The Polynomial Jacobian has n + 1 terms because of the coefficient A is composed of a unique term at the numerator. In the example above, the coefficient A is expressed as (x1*x2*x3*x4)/(x1*x2*x3*x4), and then it simplifies into taking the neutral value A = 1. One’s choosing to express the full form of the coefficient A remains readers of the homogeneous pattern is being observed among all the coefficients in the Polynomial Jacobian. In this context, the coefficient’s order of magnitude, with regards to that of the variable it multiplies in the polynomial equation for p(x), is marked as complementary to that of the variable (x). For example, the coefficient E multiplies the variable x (with the order of magnitude n = 4), but to its order of magnitude for its being complementary to that of the variable x, it must be null (ni = 0). This principle of complementarity implies that the coefficient that multiplies x⁴ only is composed of a denominator:
Ω{∏xi}ni / ∏(xi) = 1/(x1*x2*x3*x4),
with ni = 0. Indeed, its sign should be even because of the term (-1)^ni returns the value (1), which it is even sign. In summary, the expression for the Polynomial Jacobian is J(p(x)) = ∑ (-1)^ni*Ω{∏xi}ni / ∏(xi), and it represents only the coefficients that multiply the independent variable (x). Based on the complementary products of the solution set and the common denominator, all coefficients have homogeneous patterns for a given order of magnitude.
Screenshot 1:
Polynomial Jacobian for p(x) = A + Bx + Cx² + Dx³ + Ex⁴ + Fx⁵ + Gx⁶ .
Finally, I want to express my gratitude to the scientific community on Medium, for their extensive sharing of mathematic advice. I realize prolific mathematicians are producing proofs along with their results. Otherwise, I produced this article with my betting on requirement for a consensus on this article matter. Because of the sum of its parts contains novel ideas, which without any single master proof in hand, it will remain extraordinarily satisfying to readers. Thanks for reading me!
Author: Kevin Genest
