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University of Illinois – Urbana-Champaign

1. Duarte Gelvez, Eliana Maria. Syzygies and implicitization of tensor product surfaces.

Degree: PhD, Mathematics, 2017, University of Illinois – Urbana-Champaign

A tensor product surface is the closure of the image of a rational map λ : P1 ×P1 – >P3. These surfaces arise in geometric modeling and in this context it is useful to know the implicit equation of λ in P3. Currently, syzygies and Rees algebras provide the fastest and most versatile method to find implicit equations of parameterized surfaces. Knowing the structure of the syzygies of the polynomials that define the map λ allows us to formulate faster algorithms for implicitization of these surfaces and also to understand their singularities. We show that for tensor product surfaces without basepoints, the existence of a linear syzygy imposes strong conditions on the structure of the syzygies that determine the implicit equation. For tensor product surfaces with basepoints we show that the syzygies that determine the implicit equation of λ are closely related to the geometry of the set of points at which λ is undefined. Advisors/Committee Members: Schenck, Henry (advisor), Nevins, Thomas (Committee Chair), Francis, George (committee member), Reznick, Bruce (committee member).

Subjects/Keywords: Implicitization; Syzygy; Rees algebras; Basepoints; Tensor product surface; Smooth quadric

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Duarte Gelvez, E. M. (2017). Syzygies and implicitization of tensor product surfaces. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/97285

Chicago Manual of Style (16th Edition):

Duarte Gelvez, Eliana Maria. “Syzygies and implicitization of tensor product surfaces.” 2017. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed August 14, 2020. http://hdl.handle.net/2142/97285.

MLA Handbook (7th Edition):

Duarte Gelvez, Eliana Maria. “Syzygies and implicitization of tensor product surfaces.” 2017. Web. 14 Aug 2020.

Vancouver:

Duarte Gelvez EM. Syzygies and implicitization of tensor product surfaces. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2017. [cited 2020 Aug 14]. Available from: http://hdl.handle.net/2142/97285.

Council of Science Editors:

Duarte Gelvez EM. Syzygies and implicitization of tensor product surfaces. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2017. Available from: http://hdl.handle.net/2142/97285

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