Optimal Bounds for the Number of Pieces of Near-Circuit Hypersurfaces
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915153824448512 |
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| author | Deng, Weixun Rojas, J. Maurice Russell, Cordelia |
| author_facet | Deng, Weixun Rojas, J. Maurice Russell, Cordelia |
| contents | Suppose $f$ is a polynomial in $n$ variables with real coefficients, exactly $n+k$ monomial terms, and Newton polytope of positive volume. Estimating the number of connected components of the positive zero set of $f$ is a fundamental problem in real algebraic geometry, with applications in computational complexity and topology. We prove that the number of connected components is at most $3$ when $k\!=\!3$, settling an open question from Fewnomial Theory. Our results also extend to exponential sums with real exponents. A key contribution here is a deeper analysis of the underlying $\mathcal{A}$-discriminant curves, which should be of use for other quantitative geometric problems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_10590 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal Bounds for the Number of Pieces of Near-Circuit Hypersurfaces Deng, Weixun Rojas, J. Maurice Russell, Cordelia Algebraic Geometry Suppose $f$ is a polynomial in $n$ variables with real coefficients, exactly $n+k$ monomial terms, and Newton polytope of positive volume. Estimating the number of connected components of the positive zero set of $f$ is a fundamental problem in real algebraic geometry, with applications in computational complexity and topology. We prove that the number of connected components is at most $3$ when $k\!=\!3$, settling an open question from Fewnomial Theory. Our results also extend to exponential sums with real exponents. A key contribution here is a deeper analysis of the underlying $\mathcal{A}$-discriminant curves, which should be of use for other quantitative geometric problems. |
| title | Optimal Bounds for the Number of Pieces of Near-Circuit Hypersurfaces |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2502.10590 |