On the intrinsic complexity of point finding in real singular hypersurfaces

In previous work we designed an efficient procedure that finds an algebraic sample point for each connected component of a smooth real complete intersection variety. This procedure exploits geometric properties of generic polar varieties and its complexity is intrinsic with respect to the problem. In the present paper we introduce a natural construction that allows to tackle the case of a non–smooth real hypersurface by means of a reduction to a smooth complete intersection

Associated material from Blue Beach.

<p>NSM005GF045.034, the ‘Sherm Block’ (for collector Sherman Williams) of associated front and hind limb material.</p

Pelvis Type 2.

<p>NSM014GF036.001, left ilium in medial view. <b>Abbreviations</b>: <b>ap</b>, anterior process; <b>pp</b>, posterior process; <b>rr</b>, robust ridge; <b>sac</b>, sacral rib facet.</p

List of Blue Beach specimens examined in the present study.

<p>List of Blue Beach specimens examined in the present study.</p

Humerus Types 2 and 3.

<p>YPM PU 20754, left humerus in A, dorsal; and B, ventral views, less areas still obscured by matrix. RM 20.6770 left? humerus in C, dorsal, and D, ventral, views. <b>Abbreviations</b>: <b>dp</b>, deltopectoral crest; <b>ect</b>, ectepicondyle; <b>ef</b>, entepicondylar foramen.</p

Femur Type 2 and tibia Type 1.

<p>NSM004GF045.034A-C, right and left femora (Type 2) and left tibia (Type 1) found in association. A-D, I, J, left femur in A, posterior, B, ventral, C, anterior, D, dorsal I, proximal, and J, distal views. E-H right femur in E, dorsal, F, anterior, G, ventral, and H, posterior views. Left tibia in K, flexor (ventral), and L, extensor (dorsal) views. <b>Abbreviations</b>: <b>ab</b>, adductor blade; <b>ac</b>, adductor crest; <b>cr</b>, thin anterior crest; <b>ff</b>, fibular facet; <b>fo</b>, fibular fossa; <b>ft</b>, fourth trochanter; <b>icf</b>, intercondylar fossa; <b>it</b>, internal trochanter; <b>itf</b>, intertrochanteric fossa; <b>tf</b>, tibial facet.</p

Tibia Type 2.

<p>NSM014GF036.003 (A) and NSM.014.GF.036.004 (B). Right tibiae each in anterior, flexor, posterior, and extensor views. <b>Abbreviations</b>: <b>ap</b>, anterior process; <b>cc</b>, cnemial crest.</p

Syncline at Lighthouse Point.

<p>The resistant beds here include especially productive sandstones known informally as the ‘Theta Layer’ (<b>Theta</b>) and ‘Lighthouse Sandstone’ (<b>LHSS</b>).</p

Femur Type 1.

<p>YPM PU 23550, right femur in A, posterior, B, dorsal, C, anterior, and D, ventral, E, proximal, and F, distal views. Scale bar equals 10 mm. <b>Abbreviations</b>: <b>ab</b>, adductor blade; <b>ac</b>, adductor crest; <b>ff</b>, fibular facet; <b>icf</b>, intercondylar fossa; <b>tf</b>, tibial facet.</p

Location of Blue Beach.

<p>A, map of the shore line cliff along the Avon River Estuary where Horton Bluff Formation strata are exposed (continuing into the intertidal zone); B, Oldest beds (Hurd Creek Member) at the base of the section showing the road termination on the beach; C, the cliff and intertidal zone at Lighthouse Point (Blue Beach Member), where the majority of tetrapod specimens were recovered.</p