Distance to the discriminant
We will study algebraic hyper-surfaces on the real unit sphere given by an homogeneous polynomial of degree d in n variables with the view point, rarely exploited, of Euclidian geometry using Bombieri’s scalar product and norm. This view point is mostly present in works about the topology of random hyper-surfaces [ShubSmale93, GayetWelschinger14].
Our first result (lemma 3.2 page 3.2) is a formula for the distance of a polynomial to the real discriminant , i.e. the set of polynomials with a real singularity on the sphere. This formula is given for any distance coming from a scalar product on the vector space of polynomials.
Then, we concentrate on Bombieri scalar product and its remarkable properties. For instance we establish a combinatoric formula for the scalar product of two products of linear-forms (lemma 4.2 page 4.2) which allows to give a (new ?) proof of the invariance of Bombieri’s norm by composition with the orthogonal group. These properties yield a simple formula for the distance in theorem 5.3 page 5.3 from which we deduce the following inequality:
The definition 5.2 page 5.2 classifies in two categories the ways to make a polynomial singular to realise the distance to the discriminant. Then, we show, in theorem 6.3 page 6.3, that one of the category is forbidden in the case of an extremal hyper-surfaces (i.e. with maximal Betti numbers). This implies as a corollary 6.4 (page 6.4) that the above inequality becomes an equality is that case.
The main result in this paper concerns extremal hyper-surfaces that maximise the distance to the discriminant (with ). They are very remarkable objects which enjoy properties similar to those of quadratic forms: they are linear combination of powers of linear forms where the vectors are the critical points of on corresponding to the least positive critical value of . This is corollary 7.2 page 7.2 of a similar theorem 7.1 page 7.1 for all algebraic hyper-surfaces.
The next section is devoted to homogeneous polynomials in variables. We prove that a polynomial of degree with regularly spaced roots on the unit circle is a local maximum of the distance to the discriminant among polynomials with the same norm and number of roots. We conjecture that it is a global maximum and that the polynomial of degree with regularly spaced roots on the unit circle is also a similar global maximum when . This claim is supported by the fact that we were able to prove the consequence of this together with corollary 7.2 which yields to interesting trigonometric identities that we could not find somewhere else (proposition 8.3 page 8.3).
We also obtain metric information about algebraic hyper-surfaces. First, in the case of extremal hyper-surface, we give an upper bound (theorem 9.3 page 9.3) on the length of an integral curve of the gradient of in the band where is less that the least positive critical value of . Then, a general lower bound on the size and distance between the connected components of the zero locus of (corollary 10.2 and theorem 10.3).
The last section will present experimental results among which are five extremal sextic curves far from the discriminant. These are obtained by very long running numerical optimisation (many months) some of which are not terminated.
- 1 Abstract
- 2 Notation
- 3 Distance to the real discriminant
- 4 The Bombieri norm
- 5 Distance with Bombieri norm
- 6 Application to extremal hyper-surfaces
- 7 Further from the discriminant
- 8 The univariate case
- 9 Critical band of extremal hyper-surfaces
- 10 Large components far from the discriminant
- 11 Experiments with extremal curves
- 12 Conclusion
- A Proof of the inequalities for the Bombieri norm
- B Independance of
Let be the unit sphere of . We write the usual Euclidean norm on .
We consider the vector space of homogeneous polynomials in variables of degree . Let be the dimension of this vector space, we have
Let be a scalar product on and the associated norm. We use the same notation for the scalar product and norm of as for , the context should make it clear what norm we are using.
Let be an orthonormal basis of .
For , denotes the line vector and for denotes the line vector . Let be the matrix whose lines are for .
For , let be the column vector coordinates of in the basis . We may write:
We will also use the following notation for the normal and tangent component of a vector field defined for :
In the particular case of , we write and we have Euler’s relation , which gives:
Similarly, we write for the hessian matrix of at . We have that
Hence, we can find a symmetrix matrix whose kernel contains and such that :
Geometrically, is the matrix of the linear application defined as where is the projection on the plane tangent to the unit sphere at and is the second derivative of seen as a linear application.
The matrix is always of maximal rank (i.e. of rank ) for all .
Let us prove first that is of maximal rank when the elements of are monomials with arbitrary coefficients. By symmetry, we may assume that . Thus, contains the following columns coming from the partial derivatives of for :
This proves that the lines of are linearly independent when the basis contains only monomials. Second, If for some basis where of rank less that , this would yield a linear combination with some non zero coefficients such that , implying that for any polynomial we would have , and this being independent of the basis would mean that is never of maximal rank for that . ∎
3. Distance to the real discriminant
The real discriminant of the space of polynomials of degree in variables is the set of polynomials such that there exists where and .
This can be written
As usual, the equation is redundant because of the Euler’s relation which can be written here .
Therefore, the discriminant is a union of sub-vector spaces of of codimension (given that is of maximal rank).
Let be a given polynomial in . We give a way to compute the distance between and .
We first choose and we compute the distance from to . Therefore, we look for , such that:
The first condition may be written
The second condition is equivalent to orthogonal to , which means that is a linear combination of the vectors , the columns of .
This means that there exists a column vector of size such that
Let us define
is a matrix of maximal rank with . This implies that is an symmetrical and definite matrix for all . Hence, is well defined and symmetrical.
We can now write the distance to by
The above formula, established for any , is homogeneous in . We can therefore state our first lemma:
Let be an orthornomal basis of for a given scalar product. Let be the matrix defined by:
For any homogeneous polynomial , the distance to the discriminant associated to the given scalar product is given by
4. The Bombieri norm
The above lemma can be simplified in the particular case of Bombieri norm[BBEM90]. To do so, we recall the definition and properties of Bombieri norm and scalar product.
Notation: let be a vector in and , we write:
where the index of is .
Definition 4.1 (Bombieri norm and scalar product).
The Bombieri scalar product [BBEM90] for homogeneous polynomial of degree is defined by
The Bombieri scalar product and the associated norm have the remarkable property to be invariant by the action of the orthogonal group of . It was originally introduced because it verifies the Bombieri inequalities for product of polynomials. However, we do not use this property here.
We now give a lemma establishing the invariance and a result we need later in this article:
Let and be two families of vectors of . Let us consider the two following homogeneous polynomials in :
The Bombieri scalar product of these polynomials is given by the following formula which directly relates the Bombieri scalar product of polynomials to the Euclidian one in :
When the two families are constant i.e. and , this simplifies to:
We start by developing the polynomials and . For this, we use to denote applications from to and we write the vector such that .
The Bombieri norm is invariant by composition with the orthogonal group.
Proving this corollary is just proving that the Bombieri norm does not depend upon the choice of coordinates in . The last theorem establishes this for product of linear forms that generate all polynomials. ∎
We also have the following corollary, which is a way to see the Veronese embedding in the particular case of Bombieri norm:
Let be an homogeneous polynomial of degree with variables, then we have
If we write has a linear combination of monomials, the lemma 4.2 immediately gives the result. ∎
We will use the following inequality which are proved in appendix A:
For all and all , we have:
Using the following norms:
The Euclidian norm on (for and ),
The Bombieri norm for polynomials (for )
The Frobenius norm written which is the square root of the sum of the squares of the matrix coefficients (for the Hessian ).
The spectral norm written which is the largest absolute value of the eigenvalues of the matrix (also for the Hessian ).
All this inequalities are equalities for the monomial for and by invariance for power of linear form. In this case, the Hessian matrix will have only one non null eigenvalue which implies that .
5. Distance with Bombieri norm
Here is the formulation of the lemma 3.2 in the particular case of Bombieri’s norm. It can be established from lemma 3.2, but we propose a more direct proof using the invariance by composition with the orthogonal group.
Let be an homogeneous polynomial of degree with variables. The distance to the real discriminant for the Bombieri norm is given by:
Consider . We want to compute . One can always find an element of the orthogonal group such that
where the monomials and for do not appear in .
Then, using the fact that the Bombieri norm is invariant by isometry, the fact that distinct monomials are othogonal and the fact that which implies that , we have:
We can also give an alternate formulation avoiding the decomposition of the gradient in normal and tangent components:
Using and , we compute:
The first term in (5.4) is . Hence, we have:
This motivates the following definition:
Definition 5.2 (quasi-singular points, contact polynomial, contact radius).
We will call quasi-singular points for the critical points of with norm 1 where the distance to the discriminant is reached. This means that
is a quasi-singular points iff .
A necessary condition for to be a quasi singular point of is
We will say that is a contact polynomial for at if is a quasi-singular point for , (this means that has a singularity at ) and .
When is contact polynomial for at , we will say that is a contact radius for at . A contact radius is therefore the smallest polynomial for Bombieri norm that must be added to to create a singularity.
Then, we distinguish two kinds of quasi-singular points for (their names will be explaned later):
- quasi-double points:
is quasi-double point if it is a quasi-singular point of and a critical point of on the unit sphere (i.e. satisfying ).
- quasi-cusp points:
is quasi-cups point for if it is a quasi-singular point of which is not a critical point of . In this case, is a non zero member of the kernel of .
First, using the quasi-double points, we can find a very simple inequality for the distance to the discriminant:
Let be an homogeneous polynomial of degree with variables. The distance to the real discriminant for the Bombieri norm satisfies:
The condition means that is a critical point of and our theorem means that the distance to the discriminant is less or equal to the minimal critical value of in absolute value.
We use the theorem 5.1:
Let be an homogeneous polynomial of degree with variables. Let be a quasi-singular point for . Then, the contact radius at is the polynomial
and , the contact polynomial for at , has no other singularity than and .
Moreover, when , is always a quasi double point (i.e. ).
Let us assume that has another singularity and on the unit sphere (recall that we imposed quasi-singular point to lie on the unit sphere). This means that , lying at the intersection of and .
We can therefore write , where is the contact radius at :
We necessarily have . It remains to show that this is impossible. We have:
When , the hyper-surface contains the plane with multiplicity union the plane with multiplicity one. uses that same plane with replaced by , which imposes or .
When , we will show in the study of quasi-cusp point that they exist only from degree , hence we know that we only have quasi-double points, which means that . Therefore, and become:
And again, implies or . ∎
5.1. Study of quasi-double points
Let be an homogeneous polynomial of degree with variables. Let be a quasi-double point for , meaning that we have and .
The Bombieri norm being invariant by the orthogonal group, using a rotation we can assume that and that the matrix is diagonal.
Knowing that , we can write:
with no monomial of degree in in , i.e. has valuation at least in .
Then, by theorem 5.4, the contact radius is
and the contact polynomial is
The singularity at of the variety is at least a double point (justifying the name quasi-double point) and it has no other singularities by theorem 5.4.
Next, we will reveal some constraints on the eigenvalues of the hessian matrix. For this, we consider the point
and compute which is non negative because .
The same is true for all the eigenvalues and this means that when and have the same sign then (recall that by definition ).
This study establishes the following theorem:
Let be an homogeneous polynomial of degree with variables, let be a quasi-double point for and a corresponding contact polynomial at . Then, the contact radius is
The contact polynomial has only one singularity in on which is at least a double-point.
Moreover, if is an eigenvalue of with the same sign than , then .
5.2. Study of quasi-cusp point
Let be an homogeneous polynomial of degree with variables. Let be a quasi-cusp point for , meaning that we have and .
The Bombieri norm being invariant by the orthogonal group, using a rotation we can assume that and that the matrix is diagonal and that is the direction of which is an eigenvector of .
We can write:
with , , , and no monomial of degree in , nor in .
The fact that the coefficient of is null is the condition .
Then, by theorem 5.4, the contact radius is
and the contact polynomial is
The singularity at of on the unit sphere is at least a cusp (justifying the name quasi-cusp point) and it has no other singularities by theorem 5.4.
We now use a computation similar to the previous case to reveal a constraint on . For this, we consider the point and compute which is non negative because .