A moments problem II

Definitions and mathematical statement

For a positive integer $k$, let $\mathcal P_k$ consist of complex polynomials in one variable with at most $k$ nonzero monomial terms, including the zero polynomial. A nonzero member can be written \[ f(x)=\sum_{j=1}^r c_jx^{d_j},\qquad r\le k,\quad 0\le d_1<\cdots<d_r,\quad c_j\in\mathbb C\setminus\{0\}. \] The exponents are integers and have no prescribed upper bound. Define $M_m(f)=\int_0^1 f(x)^m\,dx$ for $m\ge1$, without conjugating $f$. The question is whether there exists a finite integer $N(k)$ such that \[ f\in\mathcal P_k,\quad M_1(f)=\cdots=M_{N(k)}(f)=0 \quad\Longrightarrow\quad f\equiv0. \] If such integers exist, determine the least possible values or give useful explicit bounds in terms of $k$ alone.

The essential uniformity is over all exponent choices as well as all complex coefficients. A bound depending on the largest degree would not answer the question. Using "at most $k$" includes lower-term degenerations and gives the monotone version of the source's $k$-monomial question. The known fact that vanishing of every moment forces a one-variable polynomial to be zero does not by itself supply a finite bound independent of its degree. For real coefficients the second moment suffices; the problem specifically includes complex cancellation.

Short English statement

Can finitely many moments detect every nonzero complex polynomial with $k$ terms, independently of its degree?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4700027 (AMR-046-0027), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Gasull - Some open problems in low dimensional dynamical systems (2020), Problem environment 27: A moments problem II. Original source indexed by AMR-046-0027. https://arxiv.org/abs/2012.02524.