Authorities say they are looking into all possibilities, including whether the killings were gang-related.
But they also say the timing and location of the shootings add weight to a theory that they were revenge killings carried out by the police.
The slayings occurred between Sunday night and Monday morning, just hours after an off-duty military police officer, Arides Luiz dos Santos, was killed in an attempted robbery at a gas station.
According to Brazilian media, in one of the cases, a car pulled up, its occupants told children to go inside and then opened fire.
Hundreds of angry residents took to the streets to protest the killings. They attacked buses, burning three of them.
Police have stepped up their presence in Campinas, located about 50 miles northwest of Sao Paulo, to prevent further violence.
The latest deadly incidents come as Brazil prepares to host the 2014 World Cup in June.
None of the venues for the soccer tournament is in Campinas. However, fans from around the world are expected to flock to nearby Sao Paulo, the country’s financial and business hub.
Fernando Grella Vieira, the secretary of public safety for Sao Paulo, said Tuesday that the civil and military police “will not rest” until they have found those responsible for the 13 deaths in Campinas.
No possibility is being ruled out during the investigation, he said on a visit to the city. “These are intolerable crimes and I can guarantee that the police will not rest until we solve these deaths and arrest the perpetrators,” he added.
Grella said it was too early to comment on suspicions raised by witnesses of the alleged involvement of military officers in 12 of the killings. But he stressed that he does not tolerate abuse or crimes committed by state agents.
Nine of the victims died in two separate incidents, according to the state-run Agencia Brasil. Hooded men reportedly used pistols to shoot the victims, aged between 17 and 30, in the head.
The remaining three were killed in separate attacks, the news agency said.
At least six of the victims had a criminal background, with offenses including drug trafficking, murder and auto theft, the news agency cited Licurgo Costa, director of the Judiciary Police department, as saying.
Victims ‘hit in the head, chest or abdomen’
The chief of the Civil Police’s homicide division, Devanir Dutra, told CNN affiliate TV Globo that the police will take accounts from at least 20 witnesses beginning Wednesday. “Let’s start with the relatives and friends of the victims,” he said.
According to Dutra, two people were injured and remain hospitalized in Campinas. They may be summoned by the police as witnesses.
Dutra said that the number of shots fired signaled executions. “They (the victims) were hit in the head, chest or abdomen,” he said.
One hypothesis being examined by the Civil Police is rivalry between drugs gangs, he said.