Talk:Hypersurface

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Generic title[edit]

Someone has proposed merging the energy landscape entry with hypersurface. I disagree with this proposal. While energy landscapes are a case of hypersurfaces, they have a specific physical interpretation. -- KarlHallowell 19:00, 14 May 2006 (UTC)[reply]

Specific meaning...[edit]

I agree that it has a specific meaning so it should stay as energy landscape.

Animated plot[edit]

I propose a discussion on retaining the animated plot of a single-valued function of three variables, where time is the 3rd variable. D.Lazard suggested removing it because it appears unrelated to the article, and because its "quick motion disturbs reading". Thank you.

Simiprof (talk) 19:25, 15 November 2017 (UTC)[reply]

I have notified the Wikiproject Mathematics of this discussion (WT:WPM#Animation in Hypersurface).

The reasons of my removing are well summarized in the previous post. Moreover, a function of three variable is not a hypersurface, it is only its graph which is a hypersurface. Not all hypersurfaces are graphs of functions. Also it should be unclear for most readers that the motion of the animation represents a fourth dimension. Also, the colored projection has nothing to do with the subject of the article and adds to the confusion. Thus this animation is disturbing and confusing, and does not belong to this article. D.Lazard (talk) 22:27, 15 November 2017 (UTC)[reply]

I agree that the plot is misleading and is also visually disturbing. I don't think "animation" is needed since a hypersurface is static (unlike procedure or algorithm). Can we have some static pic? like some algebraic surface in 3-dim space instead? -- Taku (talk) 23:28, 15 November 2017 (UTC)[reply]

Oppose to retaining. It is not purely subjective to be annoyed by moving, blinking, ..., simply changing objects alongside text, intended to be read. This is a well known fact about perception. Exploiting the time-"dimension" (with opposite characteristic?) this way seems wrong to me. I agree with the arguments against, cited by Simiprof and raised by D.Lazard. May your tags blink in eternity! Purgy (talk) 07:30, 16 November 2017 (UTC)[reply]
Oppose I appreciate the work that went into creating this animation and as a physicist, using time as a strategy to visualize an extra dimension feels natural to me. But I have to agree, the animation is visually distracting and temporal or a dynamic presentation has nothing to do with the general concept of hypersurface and could be confusing to reader new to the subject. Some kind of projection to lower dimensions, or maybe isosurfaces, are more typical means to visualize hypersurfaces. Also, Ackley's function is a pretty specialized kind of test surface useful in evolutionary computation or optimization, but not at all discussed in general geometry or algebraic geometry contexts.--Mark viking (talk) 13:23, 17 November 2017 (UTC)[reply]

Singularities allowed?[edit]

"A hypersurface may have singularities, and hence is not necessarily a submanifold" — Oops... if so, then this paragraph fails:

"In Rⁿ, every closed hypersurface is orientable. Every connected compact hypersurface is a level set, and separates Rⁿ into two connected components, which is related to the Jordan–Brouwer separation theorem."

A problem... Boris Tsirelson (talk) 13:26, 19 November 2017 (UTC)[reply]

I have fixed this before reading this post. In fact reference [2] has explicitly "smooth" in its title. The classical counter example is Klein bottle. D.Lazard (talk) 14:05, 19 November 2017 (UTC)[reply]

Nice. Boris Tsirelson (talk) 14:44, 19 November 2017 (UTC)[reply]

A naive question: what about $x_{1}^{2}+\dots +x_{n}^{2}=0$ ? A single point, is it an algebraic hypersurface? Boris Tsirelson (talk) 06:08, 20 November 2017 (UTC)[reply]

Yes, it is an algebraic hypersurface that has a single real point, but infinitely many complex points. Also

x_{1}^{2}+\dots +x_{n}^{2}+1=0

is an hypersurface without any real point. I plan to rewrite the section for better considering the field of coefficients, and the space(s) to which the the points belong. In particular, one talks often of "hypersurface over a field of finite characteristic". D.Lazard (talk) 10:03, 20 November 2017 (UTC)[reply]

I'll be waiting for a definition... Boris Tsirelson (talk) 10:52, 20 November 2017 (UTC)[reply]

Now, another naive question: does this set of points in $K^{n}$ determine the polynomial $P$ up to a coefficient? Probably it does for $k=\mathbb {R} ,$ $K=\mathbb {C} .$ Really? Also when $K$ is the algebraic closure of $k,$ or not so? If it does not, then, what is called an algebraic hypersurface: the set of points, or the polynomial up to a coefficient? Boris Tsirelson (talk) 18:25, 20 November 2017 (UTC)[reply]

The question is not naive, and the answer is yes if one considers points over an algebraic closed field. This is a corollary of Hilbert Nullstellensatz. I'll expand this answer in the article. D.Lazard (talk) 14:01, 21 November 2017 (UTC)[reply]

Yes, I see: "

K

is an algebraically closed extension of

k

". Following the link I am redirected to "Algebraically closed field". There I do not see "algebraically closed extension", but I see "Every field has some extension which is algebraically closed" (and a bit later, the algebraic closure). So I understand that

K

need not be the algebraic closure of

k

. Hope I am not mistaken. Boris Tsirelson (talk) 19:12, 21 November 2017 (UTC)[reply]

I have edited the redirect and its target for having the lacking definition. The field

K

may be the algebraic closure of

k

, but not necessarily. For example, if

k

is the field of rationals, one takes generally for

K

the field of complex numbers. Moreover, for technical reasons (the definition of a generic point), one takes generally for

K

an algebraically closed field that has an infinite transcendence degree over the prime field (or, equivalently, over the smallest field that contains all the coefficients of all polynomials that are considered—the number of these coefficients is necessarily finite). D.Lazard (talk) 21:19, 21 November 2017 (UTC)[reply]

Nice; thank you. Boris Tsirelson (talk) 05:46, 22 November 2017 (UTC)[reply]

Now we have two notions, "smooth" and "algebraic", which leads to a naive but maybe hard "interdisciplinary" question about their interplay. You mentioned Klein bottle. How do we know it exists as an algebraic surface, not only as a smooth immersion? And generally, can we approximate a smooth immersion (say, of a connected compact 2-dim smooth manifold to $\mathbb {R} ^{3}$ ) by algebraic surfaces (somewhat in the spirit of Weierstrass approximation theorem)? Boris Tsirelson (talk) 06:10, 22 November 2017 (UTC)[reply]

There are more notions, such as k-differentiable hypersurfaces and fractal hypersurfaces. However this article cannot repeat for hypersurfaces everything that can be said on general manifolds and algebraic varieties. We must emphasize on what is specific to hypersurfaces. I an not expert about approximations. However Stone–Weierstrass theorem asserts that differentiable multivariate functions may be approximated by polynomial. I do not believe that this implies that the zero-set of a differentiable function may be correctly approximated by an algebraic hypersurface. Another classical approach is the local study of hypersurface singularities. See for example Ak singularity. As far as I know, this theory has significative results only for isolated singularities. In any case, I am unable to expand this article in these directions. D.Lazard (talk) 11:35, 22 November 2017 (UTC)[reply]

I see. Thank you again. Boris Tsirelson (talk) 19:32, 22 November 2017 (UTC)[reply]

The matter is not at all orphaned; here are sources:

John Nash 1952 "Real algebraic manifolds" Ann. Math. 56:3 405-
René Thom 1955(?) "Approximation algébrique des applications differentiables"
Andrew Wallace 1957 "Algebraic approximation of manifolds" Proc. London Math. Soc. (3) 7 196-210
Andrew Wallace 1958 "Algebraic approximation of curves" Canad. J. Math. 10 242-278
Selman Akbulut and Henry King 1990 "Some new results on the topology of nonsingular real algebraic sets" Bull. AMS 23:2 441-
János Kollár 2001 "Which are the simplest algebraic varieties?" Bull. AMS 38:4 409-433

Boris Tsirelson (talk) 09:37, 23 November 2017 (UTC)[reply]

Who is able to expand this article in this direction? Maybe Ozob? Boris Tsirelson (talk) 06:17, 6 December 2017 (UTC)[reply]

I believe the problem with the article as it stands is that it tries to do too much at once. Suppose one fixes a geometric theory: Topological manifolds, PL manifolds, C^k-manifolds, smooth manifolds, analytic manifolds, complex manifolds, complex algebraic spaces, semi-algebraic sets, subanalytic sets, schemes, rigid analytic spaces, adic spaces, locally ringed spaces, topoi, .... In each of these theories there is a definition of a subspace, and in most of them, one can also define "codimension one". Regardless of the theory, a hypersurface is a subspace of codimension one. However, changing the theory changes the meaning of hypersurface. For instance, if C is considered as a complex manifold, then the origin is a hypersurface; but if it is considered as a real manifold, then the origin is not. The article does not try to make this distinction. It proceeds as though there were a single, universally applicable notion of a hypersurface.

Regarding the approximation question, I believe the answer is yes. A smooth manifold has a natural PL structure. A PL structure implies the existence of a semi-algebraic structure. So, locally around each point, a smooth manifold is a codimension one semi-algebraic set. I think that a codimension one semi-algebraic set is locally defined by a single equation, just as for algebraic varieties in affine or projective space. So locally in the Euclidean topology, a smooth manifold is an algebraic variety. I suppose that, if the details could be filled in, this strengthens Whitney's theorem that every smooth manifold is also an analytic manifold. (It might even be implied by Whitney's theorem and the Artin approximation theorem.)

...I looked up the answer. Bochnak, Coste, Roy, Real Algebraic Geometry, Theorem 12.4.9, says: "Given a compact connected C^∞ manifold M of dimension ≥ 2, the following conditions are equivalent: (i) M has an algebraic model X with V¹_alg(X) = 0 and R(X) is factorial; (ii) M is orientable." Here, V¹_alg(X) is the group of algebraic vector bundles, and R(X) is the coordinate ring of X. Ozob (talk) 03:55, 7 December 2017 (UTC)[reply]

So, the notion "hypersurface" is as multi-facet as "space". Wow! Then probably the article should treat first, in detail, the most notable and elementary cases, and then, shortly, more advanced cases, right? Boris Tsirelson (talk) 06:10, 7 December 2017 (UTC)[reply]

It was in this spirit that I have separated the manifold case and the algebraic case. On the other hand, some care is needed: Whitney umbrella is an algebraic real hypersurface that, in the real space is not an hypersurface. Also, an algebraic set of codimension one may be considered as an hypersurface only if all its components have dimension one. Otherwise, several equations are needed for defining it. D.Lazard (talk) 08:51, 7 December 2017 (UTC)[reply]