Minimal Polynomial Computation for Radical Expressions
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Solution Overview
Problem
Conventional methods for computing minimal polynomials for radicals over the ring Z of integer numbers or the field Q of rational numbers are inefficient, as they rely on finding annihilation polynomials and factoring, which can be cumbersome and time-consuming.
Innovation Solution
The method involves processing radicals into a level permutation set, approximating elements to find an annihilation polynomial, factoring it, and selecting the minimal polynomial by iterating with higher precision if necessary to locate a factor with a leading coefficient less than one.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If conventional methods are used to find annihilation polynomial and factor it, then minimal polynomial can be determined, but the computational process is cumbersome and time-consuming
Solution Approach 1:
The patent segments the computation process into distinct phases: (1) constructing the level permutation set from radical levels, (2) approximating elements to find the annihilation polynomial, (3) factoring the polynomial, and (4) selecting the minimal polynomial by checking leading coefficients. This segmentation allows each phase to be optimized independently, reducing overall computation time while maintaining accuracy.
Solution Approach 2:
The patent performs preliminary actions by first constructing the level permutation set and approximating elements before attempting to find the minimal polynomial. This preliminary approximation phase prepares the data in a form that facilitates faster and more accurate factorization and selection of the minimal polynomial, avoiding redundant computations.
2Measurement precision
If higher precision iteration is performed to find factor with leading coefficient less than one, then minimal polynomial is accurately identified, but computational complexity increases
Solution Approach 1:
The patent implements a feedback mechanism where the leading coefficient of each factor is checked after factorization. If a factor with leading coefficient less than one is found, the process terminates successfully. If not, the system iterates with higher precision approximation. This feedback loop ensures accurate identification of the minimal polynomial while avoiding unnecessary complexity by stopping as soon as the condition is met.
Solution Approach 2:
The patent changes the precision parameter iteratively. It starts with a base precision level for approximation, and only increases precision when needed (when no factor with leading coefficient less than one is found). This dynamic parameter adjustment balances computational complexity with the need for accurate polynomial identification.
Data Source
AI summary
Described is a technology, such as implemented in a computational software program, by which a minimal polynomial is efficiently determined for a radical expression over the ring Z of integer numbers or the ring Q of rational numbers. The levels of the radical are grouped into a level permutation group that is used to find a level permutation set. An annihilation polynomial is found based upon the level permutation set. The annihilation polynomial is factored, and a selection mechanism selects the minimal polynomial based upon the annihilation polynomial's factors.