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Related Concept Videos

Real Zeros of Polynomials01:27

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Depolymerizable Olefinic Polymers Based on Fused-Ring Cyclooctene Monomers
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Depolymerizable Olefinic Polymers Based on Fused-Ring Cyclooctene Monomers

Published on: December 16, 2022

On the Waring problem for polynomial rings.

Ralf Fröberg1, Giorgio Ottaviani, Boris Shapiro

  • 1Department of Mathematics, Stockholm University, SE-106 91, Stockholm, Sweden.

Proceedings of the National Academy of Sciences of the United States of America
|March 31, 2012
PubMed
Summary
This summary is machine-generated.

This study explores the Waring problem for polynomials. It demonstrates that homogeneous polynomials of degree k can be expressed as a sum of k(n) k-th powers of other polynomials.

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Last Updated: May 23, 2026

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Depolymerizable Olefinic Polymers Based on Fused-Ring Cyclooctene Monomers

Published on: December 16, 2022

Area of Science:

  • Algebraic Geometry
  • Commutative Algebra
  • Number Theory

Background:

  • The classical Waring problem concerns the number of k-th powers needed to represent natural numbers.
  • Extending Waring-type problems to polynomial rings is a natural progression in algebraic research.

Purpose of the Study:

  • To investigate an analog of the Waring problem for the ring of polynomials in multiple variables, C[x0, x1, ..., x(n)].
  • To determine the maximum number of k-th powers of homogeneous polynomials required to represent any general homogeneous polynomial of degree divisible by k.

Main Methods:

  • The study employs techniques from algebraic geometry and commutative algebra.
  • It involves analyzing the structure of homogeneous polynomials and their representations.

Main Results:

  • A general homogeneous polynomial p in C[x0, x1, ..., x(n)] with degree divisible by k (where k ≥ 2) can be written as a sum of at most k(n) k-th powers of homogeneous polynomials.
  • The bound k(n) is shown to be equal to the dimension count of the polynomial space.

Conclusions:

  • The research establishes a significant result for the Waring problem in the context of polynomial rings.
  • The findings provide a precise bound for the representation of homogeneous polynomials as sums of powers, aligning with dimensional analysis.