1,755 research outputs found
Real radical initial ideals
We explore the consequences of an ideal I of real polynomials having a real
radical initial ideal, both for the geometry of the real variety of I and as an
application to sums of squares representations of polynomials. We show that if
in_w(I) is real radical for a vector w in the tropical variety, then w is in
the logarithmic set of the real variety. We also give algebraic sufficient
conditions for w to be in the logarithmic limit set of a more general
semialgebraic set. If in addition the entries of w are positive, then the
corresponding quadratic module is stable. In particular, if in_w(I) is real
radical for some positive vector w then the set of sums of squares modulo I is
stable. This provides a method for checking the conditions for stability given
by Powers and Scheiderer.Comment: 16 pages, added examples, minor revision
Equations solvable by radicals in a uniquely divisible group
We study equations in groups G with unique m-th roots for each positive
integer m. A word equation in two letters is an expression of the form w(X,A) =
B, where w is a finite word in the alphabet {X,A}. We think of A,B in G as
fixed coefficients, and X in G as the unknown. Certain word equations, such as
XAXAX=B, have solutions in terms of radicals, while others such as XXAX = B do
not. We obtain the first known infinite families of word equations not solvable
by radicals, and conjecture a complete classification. To a word w we associate
a polynomial P_w in Z[x,y] in two commuting variables, which factors whenever w
is a composition of smaller words. We prove that if P_w(x^2,y^2) has an
absolutely irreducible factor in Z[x,y], then the equation w(X,A)=B is not
solvable in terms of radicals.Comment: 18 pages, added Lemma 5.2. To appear in Bull. Lon. Math. So
Maximum Likelihood for Matrices with Rank Constraints
Maximum likelihood estimation is a fundamental optimization problem in
statistics. We study this problem on manifolds of matrices with bounded rank.
These represent mixtures of distributions of two independent discrete random
variables. We determine the maximum likelihood degree for a range of
determinantal varieties, and we apply numerical algebraic geometry to compute
all critical points of their likelihood functions. This led to the discovery of
maximum likelihood duality between matrices of complementary ranks, a result
proved subsequently by Draisma and Rodriguez.Comment: 22 pages, 1 figur
Quartic Curves and Their Bitangents
A smooth quartic curve in the complex projective plane has 36 inequivalent
representations as a symmetric determinant of linear forms and 63
representations as a sum of three squares. These correspond to Cayley octads
and Steiner complexes respectively. We present exact algorithms for computing
these objects from the 28 bitangents. This expresses Vinnikov quartics as
spectrahedra and positive quartics as Gram matrices. We explore the geometry of
Gram spectrahedra and we find equations for the variety of Cayley octads.
Interwoven is an exposition of much of the 19th century theory of plane
quartics.Comment: 26 pages, 3 figures, added references, fixed theorems 4.3 and 7.8,
other minor change
Minimum-weight triangulation is NP-hard
A triangulation of a planar point set S is a maximal plane straight-line
graph with vertex set S. In the minimum-weight triangulation (MWT) problem, we
are looking for a triangulation of a given point set that minimizes the sum of
the edge lengths. We prove that the decision version of this problem is
NP-hard. We use a reduction from PLANAR-1-IN-3-SAT. The correct working of the
gadgets is established with computer assistance, using dynamic programming on
polygonal faces, as well as the beta-skeleton heuristic to certify that certain
edges belong to the minimum-weight triangulation.Comment: 45 pages (including a technical appendix of 13 pages), 28 figures.
This revision contains a few improvements in the expositio
A computer algebra user interface manifesto
Many computer algebra systems have more than 1000 built-in functions, making
expertise difficult. Using mock dialog boxes, this article describes a proposed
interactive general-purpose wizard for organizing optional transformations and
allowing easy fine grain control over the form of the result even by amateurs.
This wizard integrates ideas including:
* flexible subexpression selection;
* complete control over the ordering of variables and commutative operands,
with well-chosen defaults;
* interleaving the choice of successively less main variables with applicable
function choices to provide detailed control without incurring a combinatorial
number of applicable alternatives at any one level;
* quick applicability tests to reduce the listing of inapplicable
transformations;
* using an organizing principle to order the alternatives in a helpful
manner;
* labeling quickly-computed alternatives in dialog boxes with a preview of
their results,
* using ellipsis elisions if necessary or helpful;
* allowing the user to retreat from a sequence of choices to explore other
branches of the tree of alternatives or to return quickly to branches already
visited;
* allowing the user to accumulate more than one of the alternative forms;
* integrating direct manipulation into the wizard; and
* supporting not only the usual input-result pair mode, but also the useful
alternative derivational and in situ replacement modes in a unified window.Comment: 38 pages, 12 figures, to be published in Communications in Computer
Algebr
Implementing the asymptotically fast version of the elliptic curve primality proving algorithm
The elliptic curve primality proving (ECPP) algorithm is one of the current
fastest practical algorithms for proving the primality of large numbers. Its
running time cannot be proven rigorously, but heuristic arguments show that it
should run in time O ((log N)^5) to prove the primality of N. An asymptotically
fast version of it, attributed to J. O. Shallit, runs in time O ((log N)^4).
The aim of this article is to describe this version in more details, leading to
actual implementations able to handle numbers with several thousands of decimal
digits
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