Lower Bounds for Sums of Powers of Low Degree Univariates

Kayal, Neeraj; Koiran, Pascal; Pecatte, Timothée; Saha, Chandan

doi:10.1007/978-3-662-47672-7_66

Cited by 9 publications

(14 citation statements)

References 18 publications

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“…Lower bounds for the representation of univariate polynomials as sums of powers of low degree polynomials were recently obtained in [11]. We continue this line of work by focusing on powers of degree one polynomials.…”

Section: Motivation and Connection To Previous Workmentioning

confidence: 90%

“…Previous lower bounds for this problem [11] were only of order Ω( √ d). The polynomials in our first family are of the form H 1 (x) = k i=1 α i (x + x i ) d with all α i nonzero and the x i 's distinct.…”

Section: Introductionmentioning

confidence: 92%

“…We continue this line of work by focusing on powers of degree one polynomials. This problem is still challenging because the exponents e i may be different from d = deg(f ), and may be possibly larger than d. The lower bounds obtained in [11] are of order Ω( √ d). We obtain Ω(d) lower bounds with a new method based on polynomial interpolation (more on this below).…”

Section: Motivation and Connection To Previous Workmentioning

confidence: 99%

“…The work in [11] and in the present paper is motivated by recent progress in arithmetic circuit complexity. It was shown that strong enough lower bounds for circuits of depth four [1,14,19] or even depth three [9,19] would yield a separation of Valiant's [20] algebraic complexity classes VP and VNP.…”

Section: Motivation and Connection To Previous Workmentioning

confidence: 99%

“…It is therefore desirable to develop new lower bounds methods. We view the models studied in [11] and in the present paper as "test beds" for the development of such methods in a fairly simple setting. We note also that (as explained above) strong lower bounds in slightly more general models would imply a separation of VP from VNP.…”

Section: Motivation and Connection To Previous Workmentioning

confidence: 99%

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Lower bounds by Birkhoff interpolation

Garca-Marco

Koiran

2017

Journal of Complexity

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In this paper we give lower bounds for the representation of real univariate polynomials as sums of powers of degree 1 polynomials. We present two families of polynomials of degree d such that the number of powers that are required in such a representation must be at least of order d. This is clearly optimal up to a constant factor. Previous lower bounds for this problem were only of order Ω( √ d), and were obtained from arguments based on Wronskian determinants and "shifted derivatives." We obtain this improvement thanks to a new lower bound method based on Birkhoff interpolation (also known as "lacunary polynomial interpolation").

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Section: Motivation and Connection To Previous Workmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 92%

Section: Motivation and Connection To Previous Workmentioning

confidence: 99%

Section: Motivation and Connection To Previous Workmentioning

confidence: 99%

Section: Motivation and Connection To Previous Workmentioning

confidence: 99%

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Lower bounds by Birkhoff interpolation

Garca-Marco

Koiran

2017

Journal of Complexity

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Sums of Read-Once Formulas: How Many Summands Suffice?

Mahajan

Tawari

2016

Computer Science – Theory and Applications

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Weighted Sum-of-Squares Lower Bounds for Univariate Polynomials Imply $$\text{VP} \neq \text{VNP}$$

Dutta,

Saxena,

Thierauf

2024

comput. complex.

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For a polynomial f, a weighted sum-of-squares representation (SOS) has the form $$f = \sum_{i\in [s]} c_i f_i^2$$ f = ∑ i ∈ [ s ] c i f i 2 , where the weights$$c_i$$ c i are field elements. The size of the representation is the number of monomials that appear across the $$f_i$$ f i 's. Its minimum across all such decompositions is called the support-sum S(f) of f.For a univariate polynomial f of degree d of full support, a lower bound for the support-sum is $$S(f) \ge \sqrt d$$ S ( f ) ≥ d . We show that the existence of an explicit univariate polynomial f with support-sum just slightly larger than the lower bound, that is, $$S(f) \ge d^{0.5+\varepsilon}$$ S ( f ) ≥ d 0.5 + ε , for some $$\varepsilon > 0$$ ε > 0 , implies that $$\ne$$ ≠ , the major open problem in algebraic complexity. In fact, our proof works for some subconstant functions $$\varepsilon(d) > 0$$ ε ( d ) > 0 as well. We also consider the sum-of-cubes representation (SOC) of polynomials. We show that an explicit hard polynomial implies both blackbox-PIT is in , and $$\neq$$ ≠ .

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Lower Bounds for Sums of Powers of Low Degree Univariates

Cited by 9 publications

References 18 publications

Lower bounds by Birkhoff interpolation

Lower bounds by Birkhoff interpolation

Sums of Read-Once Formulas: How Many Summands Suffice?

Weighted Sum-of-Squares Lower Bounds for Univariate Polynomials Imply $$\text{VP} \neq \text{VNP}$$

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