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NP-hardness of testing equivalence to sparse polynomials and to constant-support polynomials

dc.contributor.authorBaraskar, Omkar Bhalchandra
dc.date.accessioned2025-07-22T19:16:02Z
dc.date.available2025-07-22T19:16:02Z
dc.date.issued2025-07-22
dc.date.submitted2025-07-17
dc.description.abstractGiven a list of monomials of a n-variate polynomial f over a field F, and an integer s, decide whether there exists an invertible transform A and a b such that f(Ax + b) has less than s monomials. This problem is called the Equivalence testing to sparse polynomials (ETsparse). It was studied in [GrigorievK93] over Q, in this work, they give an exponential in n^4 time algorithm for the problem. The lack of progress in the complexity of the problem over last three decades raises a question, is ETsparse hard? In this thesis we give an affirmative answer to the question by showing that it is NP-hard over any field. Sparse orbit complexity of a polynomial f is the smallest integer s_0 such that there exists an invertible transform A such that f(Ax) has s_0 monomials. Since ETsparse is NP-hard hence computing the sparse orbit complexity is also NP-hard. We also show that approximating the sparse orbit complexity upto a factor of s_f^{1/3-\epsilon} for any \epsilon \in (0,1/3) is NP-hard, where s_f is the number of monomials in f. Interestingly, this approximation result has been shown without invoking the celebrated PCP theorem. [ChillaraGS23] study a variant of the problem which focus on shift equivalence. More precisely, given f over some ring R (the input has the same representation as in ETsparse) and an integer s, does there exists a b such that f(x + b) has less than s monomials. It is called the SETsparse problem, [ChillaraGS23] showed that SETsparse is NP-hard when R is an integral domain which is not a field; we extend their result to the case when R is a field. Finally, we also study the problem of testing equivalence to constant-support polynomials; more precisely, given a polynomial f as before and with support \sigma, does there exists an invertible transform A such that f(Ax) has support \sigma -1. We call this problem ETsupport. We show that ETsupport is NP-hard for \sigma >= 5 and over any field.
dc.identifier.urihttps://hdl.handle.net/10012/22043
dc.language.isoen
dc.pendingfalse
dc.publisherUniversity of Waterlooen
dc.subjectcomplexity theory
dc.subjectalgebraic complexity
dc.subjectequivalence testing
dc.titleNP-hardness of testing equivalence to sparse polynomials and to constant-support polynomials
dc.typeMaster Thesis
uws-etd.degreeMaster of Mathematics
uws-etd.degree.departmentDavid R. Cheriton School of Computer Science
uws-etd.degree.disciplineComputer Science
uws-etd.degree.grantorUniversity of Waterlooen
uws-etd.embargo.terms0
uws.contributor.advisorOliveira, Rafael
uws.contributor.advisorSchost, Eric
uws.contributor.affiliation1Faculty of Mathematics
uws.peerReviewStatusUnrevieweden
uws.published.cityWaterlooen
uws.published.countryCanadaen
uws.published.provinceOntarioen
uws.scholarLevelGraduateen
uws.typeOfResourceTexten

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