Nonclassical symmetries of polynomial equations and test problems with parameters for computer algebra systems

Technical Summary: Nonclassical Symmetries of Polynomial Equations and Test Problems for Computer Algebra Systems

Problem Statement
The paper addresses the challenge of solving polynomial equations and systems containing free parameters. While classical algebraic methods and modern Computer Algebra Systems (CAS) like Maple and Mathematica are effective for finding roots of equations with fixed numerical coefficients, they often fail to provide explicit analytical solutions (in radicals) for equations containing arbitrary parameters. The authors identify a gap in current CAS capabilities regarding the efficient discovery of radical solutions for specific high-degree polynomial equations that possess "hidden" symmetries but do not appear symmetric in their standard form.

Methodology
The authors employ a theoretical framework based on symmetry analysis and reduction techniques to transform complex polynomial equations into solvable systems. The methodology proceeds through several stages:

1. Symmetry Classification: The paper distinguishes between:

   • Classical Symmetric Systems: Systems where equations remain invariant under the permutation of variables ($x \leftrightarrow y$). These are solved by introducing elementary symmetric polynomials ($\sigma_1 = x+y, \sigma_2 = xy$).
   • Mixed-Type Systems: Systems containing one symmetric and one anti-symmetric polynomial. These are shown to split into simpler subsystems, one of which reduces to a single variable equation.
   • Nonclassical Symmetric Systems: Systems of the form $P(x, y) + Q_s(x, y) = 0$ and $P(y, x) + Q_s(x, y) = 0$, where equations swap places upon variable permutation. By adding and subtracting these equations, the system is reduced to a symmetric system and an anti-symmetric condition $(x-y)R_s(x,y)=0$.

2. Reduction via Hidden Symmetries: The core methodological contribution involves identifying equations with "hidden" symmetries. By introducing an auxiliary variable (e.g., $y = f(x)$ for equations of the form $f(f(x)) = x$), the authors demonstrate that specific high-degree polynomial equations can be transformed into the nonclassical symmetric systems described above. This reduction allows the original high-degree equation to be decomposed into lower-degree equations (cubic or quadratic) that are solvable in radicals.

3. Computational Verification: To test the limits of current technology, the authors formulate three specific "test problems" with parameters derived from their theoretical reductions:

   • A sixth-degree equation: $(a - x^2)^3 = (b - x^3)^2$.
   • A ninth-degree equation: $(x^3 + a)^3 + a = x$.
   • A ninth-degree equation: $(x^3 + x + b)^3 + x^3 + 2b = 0$.
     These problems were solved using Maple (2024) and Mathematica (13) to compare analytical and numerical performance against the known radical solutions derived in the paper.

Key Contributions

• Theoretical Reductions: The paper establishes constructive proofs that specific families of high-degree polynomial equations (sixth and ninth degree) containing free parameters can be reduced to systems solvable in radicals. This includes equations involving the second iteration of a polynomial and equations with "hidden" symmetries that are not immediately obvious.
• Decomposition of Nonclassical Systems: A systematic method is presented for solving nonclassical symmetric systems where equations change places upon variable permutation, splitting them into a symmetric subsystem and a linear/anti-symmetric condition.
• Benchmarking CAS Capabilities: The authors provide a rigorous benchmark of Maple and Mathematica. They demonstrate that while these systems can find all numerical roots for fixed parameter values, they frequently fail to express all roots in radical form when free parameters are present.
  • For the sixth-degree two-parameter equation, Maple and Mathematica fail to find explicit radical solutions for general parameters, returning results in terms of RootOf or Root functions.
  • For the ninth-degree one-parameter equations, the systems can find three roots (corresponding to a cubic factor) in radicals but fail to express the remaining six roots in radical form for general parameters.
  • The paper notes that for specific integer parameter values, Maple can sometimes find all nine roots in radicals, whereas Mathematica often finds only five, highlighting inconsistencies in their symbolic engines.

Results
The computational tests reveal significant limitations in current CAS regarding parametric polynomial equations:

• Analytical Failure: Neither Maple nor Mathematica could efficiently find the complete set of analytical solutions (in radicals) for the proposed test problems when parameters were left as free variables.
• Numerical Success: Both systems successfully computed all numerical roots (real and complex) when specific numerical values were assigned to the parameters.
• Degenerate Cases: In special cases where parameters were set to zero (degenerating the equations), both systems successfully produced radical solutions, confirming that the failure lies in handling the general parametric form rather than the specific algebraic structure of the reduced equations.

Significance
The paper claims that its primary significance lies in providing a set of "test problems with parameters" specifically designed to analyze and improve the capabilities of Computer Algebra Systems. By demonstrating that equations with known radical solutions exist but remain unsolved in radical form by leading CAS, the authors argue that these systems require further development to handle nonclassical symmetries and hidden reductions effectively. The work serves as a diagnostic tool for developers to identify gaps in symbolic solving algorithms for parametric polynomial systems.