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Part 4 (continued from: https://www.freemathhelp.com/forum/...n-to-unlock-their-secrets-iii-quantas.128431/)
www.quantamagazine.org
Counting the Overlap
In order to count the number of overlaps, DeMarco, Krieger and Ye turned to a tool which measures how much the value of an initial point grows as it’s repeatedly added to itself.
The torsion points on elliptic curves have no growth or long-term change, since they circle back to themselves. Mathematicians measure this growth, or lack of it, using a “height function.” It equals zero when applied to the torsion points of elliptic curves. Similarly, it equals zero when applied to the finite orbit points of dynamical systems. Height functions are an essential tool in arithmetic dynamics because they can be used on either side of the divide between the two branches.
The authors studied how often points of zero height coincide for the dynamical systems representing the elliptic curves. They showed that these points are sufficiently scattered around the complex plane so that they are unlikely to coincide — so unlikely, in fact, that they can’t do it more than a specific number of times.
That number is difficult to compute, and it’s probably much larger than the actual number of coinciding points, but the authors proved that this hard ceiling does exist. They then translated the problem back into the language of number theory to determine a maximum number of shared torsion points on two elliptic curves — the key to their original question and a provocative demonstration of the power of arithmetic dynamics.
“They’re able to answer a specific question that already existed just within number theory and that nobody thought had anything to do with dynamical systems,” said Patrick Ingram of York University in Toronto. “That got a lot of attention.”
RELATED:
Mathematicians Shed Light on Minimalist Conjecture
3-D Fractals Offer Clues to Complex Systems
Mathematician Measures the Repulsive Force Within Polynomials
Without a Proof, Mathematicians Wonder How Much Evidence Is Enough
Shortly after DeMarco, Krieger and Ye first posted their proof of a uniform bound for the Manin-Mumford conjecture, they released a second, related paper. The follow-up work is about a question in dynamical systems, instead of number theory, but it uses similar methods. In that sense, the pair of papers is a quintessential product of the analogy Silverman noticed almost 30 years earlier.
“In some sense, it’s the same argument applied to two different families of examples,” said DeMarco.
The two papers synthesized many of the ideas that mathematicians working in arithmetic dynamics have developed over the last three decades while also adding wholly new techniques. But Silverman sees the papers as suggestive more than conclusive, hinting at an even wider influence for the new discipline.
Mathematicians Set Numbers in Motion to Unlock Their Secrets
A new proof demonstrates the power of arithmetic dynamics, an emerging discipline that combines insights from number theory and dynamical systems.
www.quantamagazine.org
Counting the Overlap
In order to count the number of overlaps, DeMarco, Krieger and Ye turned to a tool which measures how much the value of an initial point grows as it’s repeatedly added to itself.
The torsion points on elliptic curves have no growth or long-term change, since they circle back to themselves. Mathematicians measure this growth, or lack of it, using a “height function.” It equals zero when applied to the torsion points of elliptic curves. Similarly, it equals zero when applied to the finite orbit points of dynamical systems. Height functions are an essential tool in arithmetic dynamics because they can be used on either side of the divide between the two branches.
The authors studied how often points of zero height coincide for the dynamical systems representing the elliptic curves. They showed that these points are sufficiently scattered around the complex plane so that they are unlikely to coincide — so unlikely, in fact, that they can’t do it more than a specific number of times.
That number is difficult to compute, and it’s probably much larger than the actual number of coinciding points, but the authors proved that this hard ceiling does exist. They then translated the problem back into the language of number theory to determine a maximum number of shared torsion points on two elliptic curves — the key to their original question and a provocative demonstration of the power of arithmetic dynamics.
“They’re able to answer a specific question that already existed just within number theory and that nobody thought had anything to do with dynamical systems,” said Patrick Ingram of York University in Toronto. “That got a lot of attention.”
RELATED:
Mathematicians Shed Light on Minimalist Conjecture
3-D Fractals Offer Clues to Complex Systems
Mathematician Measures the Repulsive Force Within Polynomials
Without a Proof, Mathematicians Wonder How Much Evidence Is Enough
Shortly after DeMarco, Krieger and Ye first posted their proof of a uniform bound for the Manin-Mumford conjecture, they released a second, related paper. The follow-up work is about a question in dynamical systems, instead of number theory, but it uses similar methods. In that sense, the pair of papers is a quintessential product of the analogy Silverman noticed almost 30 years earlier.
“In some sense, it’s the same argument applied to two different families of examples,” said DeMarco.
The two papers synthesized many of the ideas that mathematicians working in arithmetic dynamics have developed over the last three decades while also adding wholly new techniques. But Silverman sees the papers as suggestive more than conclusive, hinting at an even wider influence for the new discipline.