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Sums of Polynomials and Clique Roots

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Hossein Teimoori Faal

Combinatorics · Submitted 17 Dec 2021

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In this paper, pursuing the same line of ideas in the proof of an old longstanding open conjecture of \emph{Kadison-Singer} , we introduce a key lemma which we call it the interlacing lemma which indicates a necessary condition for having a real root for sums of polynomials with (at least) one real root.

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arXiv ID2112.09744 (version 1)
Pith IDBNUFOSUM, Pith's permanent record of this paper and its review
CitationHossein Teimoori Faal. "Sums of Polynomials and Clique Roots." arXiv:2112.09744 (2021). https://pith.science/paper/BNUFOSUM
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,995 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.30 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.17 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; .

In this paper, pursuing the same line of ideas in the proof of an old longstanding open conjecture of \emph{Kadison-Singer} , we introduce a key lemma which we call it the interlacing lemma which indicates a necessary condition for having a real root for sums of polynomials with (at least) one real root. Then, as an immediate application of this simple but potentially useful lemma we characterize several class of graphs which have only clique roots. Finally, we conclude our paper with several interesting open problems and conjectures for interested readers.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 In this paper, pursuing the same line of ideas in the proof of an old longstanding open conjecture of \emph{Kadison-Singer} , we introduce a key lemma which we call it the interlacing lemma which indicates a necessary condition for having a real root for sums of polynomials with (at least) one real root. contribution 0.98 0.94 0.33 asserted 0.52
s1 Then, as an immediate application of this simple but potentially useful lemma we characterize several class of graphs which have only clique roots. result 0.81 0.95 0.33 asserted 1.00
s2 Finally, we conclude our paper with several interesting open problems and conjectures for interested readers. future work 1.00 0.66 0.08 asserted 0.73

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

In this paper, pursuing the same line of ideas in the proof of an old longstanding open conjecture of \emph{Kadison-Singer} , we introduce a key lemma which we call it the interlacing lemma which indicates a necessary condition for having a real root for sums of polynomials with (at least) one real root. Then, as an immediate application of this simple but potentially useful lemma we characterize several class of graphs which have only clique roots. Finally, we conclude our paper with several interesting open problems and conjectures for interested readers.

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