Matches in SemOpenAlex for { <https://semopenalex.org/work/W58648772> ?p ?o ?g. }
Showing items 1 to 75 of
75
with 100 items per page.
- W58648772 startingPage "118" @default.
- W58648772 abstract "Let R = F[x1, x2, . . . , xn]. For a polynomial f ∈ R[y], we say that a polynomial p ∈ R is a root of f , if f(p) = 0. We study the relation between the arithmetic circuit sizes of f and p for general circuits and algebraic branching programs. An algebraic branching program (ABP) is given by a layered directed acyclic graph with source σ and sink τ , whose edges are labeled by variables or field constants. It computes the sum of weights of all paths from σ to τ , where the weight of a path is defined as the product of edge-labels on the path. For the size of an ABP we count the number of nodes in the underlying graph. We address the following fundamental question: suppose the polynomial f can be computed by an ABP of size s. Is the ABP size of every root p of f guaranteed to be bounded by a polynomial in s ? For general circuits it is known that the circuit size of any root p of a polynomial f with circuit size s is at most poly(s, deg(p),m), where m is the multiplicity of p in f , i.e. m is the largest number such that (p− y) divides f . This bound follows from a result about factors of arithmetic circuits independently obtained by Kaltofen [1] and Burgisser [2]. In this paper, we study the above question for ABPs for the case where f is assumed to factor as f = p0 · (p1 − y)(p2 − y) . . . (pr − y), for p0, p1, . . . , pr ∈ F[x1, x2, . . . , xn] with p0 6= 0 and |{p1, p2, . . . , pr}| = r, and where p1 is degree-dominant in the sense that mindeg(p1) > max2≤i≤rdeg(pi). For this situation, provided F has characteristic zero, we show that p1 can be computed by an ABP of size polynomial in s. To prove the above result, we view the question as a problem of computing eigenvalues. Roughly, the pis are made to appear as the eigenvalues of some matrix over the field F(x1, x2, . . . , xn) of rational functions. This problem is then solved by adapting the numerical method of power iteration to our situation. Using power iteration makes the computation amenable to be coded out as an ABP, since ABPs can efficiently compute iterated matrix multiplication. In this work we adapt techniques which are well-known from numerical analysis, for use in the area of arithmetic circuit complexity. Staying with this theme, we also improve the above mentioned poly(s, deg(p),m) bound for the circuit size of a root p of a polynomial f computed by an (unrestricted) arithmetic circuit of size s. Rather than applying Ref. [1, 2], we develop a discrete analogue of Newton’s Method. ∗Laboratory for Foundations of Computer Science, School of Informatics, The University of Edinburgh. Email: maurice.julien.jansen@gmail.com. This work was conducted while the author was a Postdoctoral Fellow at the Institute for Theoretical Computer Science of Tsinghua University, and it was supported in part by the National Natural Science Foundation of China Grant 60553001, 61073174, 61033001 and the National Basic Research Program of China Grant 2007CB807900, 2007CB807901." @default.
- W58648772 created "2016-06-24" @default.
- W58648772 creator A5016629535 @default.
- W58648772 date "2010-01-01" @default.
- W58648772 modified "2023-09-25" @default.
- W58648772 title "Extracting Roots of Arithmetic Circuits by Adapting Numerical Methods." @default.
- W58648772 hasPublicationYear "2010" @default.
- W58648772 type Work @default.
- W58648772 sameAs 58648772 @default.
- W58648772 citedByCount "0" @default.
- W58648772 crossrefType "journal-article" @default.
- W58648772 hasAuthorship W58648772A5016629535 @default.
- W58648772 hasConcept C114614502 @default.
- W58648772 hasConcept C118615104 @default.
- W58648772 hasConcept C132525143 @default.
- W58648772 hasConcept C134306372 @default.
- W58648772 hasConcept C156004811 @default.
- W58648772 hasConcept C159985019 @default.
- W58648772 hasConcept C192562407 @default.
- W58648772 hasConcept C199360897 @default.
- W58648772 hasConcept C206175624 @default.
- W58648772 hasConcept C2777735758 @default.
- W58648772 hasConcept C33923547 @default.
- W58648772 hasConcept C34388435 @default.
- W58648772 hasConcept C41008148 @default.
- W58648772 hasConcept C63553672 @default.
- W58648772 hasConcept C77553402 @default.
- W58648772 hasConcept C90119067 @default.
- W58648772 hasConcept C9376300 @default.
- W58648772 hasConceptScore W58648772C114614502 @default.
- W58648772 hasConceptScore W58648772C118615104 @default.
- W58648772 hasConceptScore W58648772C132525143 @default.
- W58648772 hasConceptScore W58648772C134306372 @default.
- W58648772 hasConceptScore W58648772C156004811 @default.
- W58648772 hasConceptScore W58648772C159985019 @default.
- W58648772 hasConceptScore W58648772C192562407 @default.
- W58648772 hasConceptScore W58648772C199360897 @default.
- W58648772 hasConceptScore W58648772C206175624 @default.
- W58648772 hasConceptScore W58648772C2777735758 @default.
- W58648772 hasConceptScore W58648772C33923547 @default.
- W58648772 hasConceptScore W58648772C34388435 @default.
- W58648772 hasConceptScore W58648772C41008148 @default.
- W58648772 hasConceptScore W58648772C63553672 @default.
- W58648772 hasConceptScore W58648772C77553402 @default.
- W58648772 hasConceptScore W58648772C90119067 @default.
- W58648772 hasConceptScore W58648772C9376300 @default.
- W58648772 hasLocation W586487721 @default.
- W58648772 hasOpenAccess W58648772 @default.
- W58648772 hasPrimaryLocation W586487721 @default.
- W58648772 hasRelatedWork W1837725692 @default.
- W58648772 hasRelatedWork W1988709433 @default.
- W58648772 hasRelatedWork W2000741931 @default.
- W58648772 hasRelatedWork W2019969751 @default.
- W58648772 hasRelatedWork W2140994271 @default.
- W58648772 hasRelatedWork W2217345548 @default.
- W58648772 hasRelatedWork W2399461329 @default.
- W58648772 hasRelatedWork W2592730710 @default.
- W58648772 hasRelatedWork W2610303141 @default.
- W58648772 hasRelatedWork W2803788777 @default.
- W58648772 hasRelatedWork W2953354114 @default.
- W58648772 hasRelatedWork W2954888688 @default.
- W58648772 hasRelatedWork W2963377858 @default.
- W58648772 hasRelatedWork W2963409396 @default.
- W58648772 hasRelatedWork W3038062119 @default.
- W58648772 hasRelatedWork W3044232313 @default.
- W58648772 hasRelatedWork W3094484287 @default.
- W58648772 hasRelatedWork W3112278279 @default.
- W58648772 hasRelatedWork W3164684355 @default.
- W58648772 hasRelatedWork W3183759618 @default.
- W58648772 hasVolume "17" @default.
- W58648772 isParatext "false" @default.
- W58648772 isRetracted "false" @default.
- W58648772 magId "58648772" @default.
- W58648772 workType "article" @default.