2 problems

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Nonexistence conjecture for balancing configurations with unequal degree difference
Nonexistence conjecture. When $m-n>1$ , there will be no balancing configurations satisfying those requirements.
Conjectureopenadler-moser-polynomials gross-pitaevskii-equation vortex-ringsarXiv posted 2 days ago
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The simple-root conjecture for Adler–Moser polynomials
Simple-root conjecture. The Adler–Moser polynomial $A_n$ has only simple roots for every $N$ .
Conjectureopenadler-moser-polynomials integrable-systems traveling-wavesarXiv posted 2 days ago

Nonexistence conjecture for balancing configurations with unequal degree difference