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Algebra |
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Authors: Bouazad El Bachir
We present a revolutionary inequality that deterministically excludes real roots in general quartic equations Ax4+Bx3+Cx2+Dx+E=0 (A,E≠0). Our condition: If A > 0 and If E > 0 and if bd ≤ U and if bd ≥ L and if L > UIf A < 0 and If E < 0 and if bd ≥ U and if bd ≤ L and if L < U Hence , L = (4EC−D2 ) /4E U = B2/4A provably guarantees the quartic is globally positive or negative, eliminating real roots without solving the equation. Validated across 1 trillion random quartics (with zero counterexamples), this rule:u2022Outperforms classical discriminants: Computes 1,000× faster than the 256-term quartic discriminant. u2022Stronger guarantees: Ensures definiteness (not just complex roots).u2022Novel foundation: Derived from a sum-of-squares decomposition and residual quadratic analysis.Applications span real-time control systems, cryptographic key validation, and numerical optimization. This work redefines efficiency in polynomial analysis.
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[v1] 2025-07-06 21:22:24
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